Polynomial Expressions
A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
Poly...
A mathematics expression is a calculation procedure which
is written using numbers, variables, and operation-symbols.
The ...
For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
A mathematics expression is a calcula...
For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
A mathematics expression is a calcula...
For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times...
For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times...
For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times...
For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times...
For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times...
For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times...
For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times...
For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times...
For example, let x represent a number,
then “2 + 3x” means to “sum 2 and 3 times x”,
“4x2 – 5x” means to
“subtract 5 times...
b. –3y2 (y = –4)
Polynomial Expressions
b. –3y2 (y = –4)
–3y2  –3(–4)2
Polynomial Expressions
b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
Polynomial Expressions
b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
Polynomial Expressions
b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
Polynomial Expressions
b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64)
Polynomial Expressions
b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
Polynomial Expressions
b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
Polynomial Expressions
Polynomial...
b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called p...
b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called p...
b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called p...
b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called p...
b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called p...
b. –3y2 (y = –4)
–3y2  –3(–4)2
= –3(16) = –48.
c. –3y3
–3y3  – 3(–4)3
= – 3(–64) = 192
The sum of monomials are called p...
Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
Polynomial Expressions
Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 ...
Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 ...
Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 ...
Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 ...
Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 ...
Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 ...
Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 ...
Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 ...
Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3.
The polynomial 4x2 – 3x3 is the combination of two
monomials; 4x2 ...
Each term is addressed by the variable part.
Polynomial Expressions
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2,
Polynomial Expressions
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
Polynomial...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Each term is addressed by the variable part. Hence the
x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,
and the nu...
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
Polynomial Expressions
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
Polynomial Expressions
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
Polynomial Expressions
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6...
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6...
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6...
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6...
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6...
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6...
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6...
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6...
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6...
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6...
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6...
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6...
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6...
Example C. Expand and simplify.
a. 3(2x – 4) + 2(4 – 5x)
= 6x – 12 + 8 – 10x
= –4x – 4
b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6...
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
Polynomial Expressions
= 6xy – 8x2y + 2xy – 3xy2
Polynomial Expressions
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
Polynomial Exp...
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8x...
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8x...
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8x...
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8x...
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8x...
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8x...
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8x...
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8x...
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8x...
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8x...
Example D. Expand and simplify.
a. 2(3xy – 4x2y) + 2xy – 3xy2
= 6xy – 8x2y + 2xy – 3xy2
= 8xy – 8x2y – 3xy2
b. Evaluate 8x...
Ex. A. Evaluate each monomials with the given values.
3. 2x2 with x = 1 and x = –1 4. –2x2 with x = 1 and x = –1
5. 5y3 wi...
21. x2 – 3x + 5 + 2(–x2 – 4x – 6)
22. x2 – 3x + 5 – 2(–x2 – 4x – 6)
23. 2(x2 – 3x + 5) + 5(–x2 – 4x – 6)
24. 2(x2 – 3x + 5...
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4 3polynomial expressions

  1. 1. Polynomial Expressions
  2. 2. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. Polynomial Expressions
  3. 3. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Polynomial Expressions
  4. 4. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Polynomial Expressions
  5. 5. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous.
  6. 6. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous.
  7. 7. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous.
  8. 8. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous. These are complicated!
  9. 9. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. The simplest form of expression are #xN, where N is a non- negative integer and # is a number, is called a monomial (one-term). Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous. These are complicated!
  10. 10. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. The simplest form of expression are #xN, where N is a non- negative integer and # is a number, is called a monomial (one-term). For example –1, 2x, 3x2, and –4x3 are monomials. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous. These are complicated!
  11. 11. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. The simplest form of expression are #xN, where N is a non- negative integer and # is a number, is called a monomial (one-term). For example –1, 2x, 3x2, and –4x3 are monomials. If N = 0 we’ve the constants, N = 1, the linear monomials #x. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous. These are complicated!
  12. 12. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Example A. Evaluate the monomials if y = –4 a. 3y2 The simplest form of expression are #xN, where N is a non- negative integer and # is a number, is called a monomial (one-term). For example –1, 2x, 3x2, and –4x3 are monomials. If N = 0 we’ve the constants, N = 1, the linear monomials #x. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous. These are complicated!
  13. 13. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Example A. Evaluate the monomials if y = –4 a. 3y2 3y2  3(–4)2 The simplest form of expression are #xN, where N is a non- negative integer and # is a number, is called a monomial (one-term). For example –1, 2x, 3x2, and –4x3 are monomials. If N = 0 we’ve the constants, N = 1, the linear monomials #x. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous. These are complicated!
  14. 14. For example, let x represent a number, then “2 + 3x” means to “sum 2 and 3 times x”, “4x2 – 5x” means to “subtract 5 times x from 4 times the square of x”, (3 – 2x)2 means to “square of the difference of 3 and twice x”. A mathematics expression is a calculation procedure which is written using numbers, variables, and operation-symbols. The purposes of this symbolic-form are clarity and simplicity. Example A. Evaluate the monomials if y = –4 a. 3y2 3y2  3(–4)2 = 3(16) = 48 The simplest form of expression are #xN, where N is a non- negative integer and # is a number, is called a monomial (one-term). For example –1, 2x, 3x2, and –4x3 are monomials. If N = 0 we’ve the constants, N = 1, the linear monomials #x. Polynomial Expressions The English phrase “sum 2 and 3 times x” is ambiguous. These are complicated!
  15. 15. b. –3y2 (y = –4) Polynomial Expressions
  16. 16. b. –3y2 (y = –4) –3y2  –3(–4)2 Polynomial Expressions
  17. 17. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. Polynomial Expressions
  18. 18. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 Polynomial Expressions
  19. 19. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 Polynomial Expressions
  20. 20. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) Polynomial Expressions
  21. 21. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 Polynomial Expressions
  22. 22. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 Polynomial Expressions Polynomial Expressions
  23. 23. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 The sum of monomials are called polynomials (many-terms). Polynomial Expressions Polynomial Expressions
  24. 24. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 The sum of monomials are called polynomials (many-terms). These are expressions of the form, arranged in the order of powers of the x: #xN ± #xN-1 ± … ± #x1 ± # where the #’s are numbers. Polynomial Expressions Polynomial Expressions
  25. 25. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 The sum of monomials are called polynomials (many-terms). These are expressions of the form, arranged in the order of powers of the x: #xN ± #xN-1 ± … ± #x1 ± # where the #’s are numbers. The highest exponent N is the degree of the polynomial. Polynomial Expressions Polynomial Expressions
  26. 26. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 The sum of monomials are called polynomials (many-terms). These are expressions of the form, arranged in the order of powers of the x: #xN ± #xN-1 ± … ± #x1 ± # where the #’s are numbers. The highest exponent N is the degree of the polynomial. For example, 4x – 7 is 1st degree (linear) Polynomial Expressions Polynomial Expressions
  27. 27. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 The sum of monomials are called polynomials (many-terms). These are expressions of the form, arranged in the order of powers of the x: #xN ± #xN-1 ± … ± #x1 ± # where the #’s are numbers. The highest exponent N is the degree of the polynomial. For example, 4x – 7 is 1st degree (linear) and the degree of 1 – 3x2 – πx40 is 40. Polynomial Expressions Polynomial Expressions
  28. 28. b. –3y2 (y = –4) –3y2  –3(–4)2 = –3(16) = –48. c. –3y3 –3y3  – 3(–4)3 = – 3(–64) = 192 The sum of monomials are called polynomials (many-terms). These are expressions of the form, arranged in the order of powers of the x: #xN ± #xN-1 ± … ± #x1 ± # where the #’s are numbers. The highest exponent N is the degree of the polynomial. For example, 4x – 7 is 1st degree (linear) and the degree of 1 – 3x2 – πx40 is 40. x 1 is not a polynomial.The expression Polynomial Expressions Polynomial Expressions
  29. 29. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. Polynomial Expressions
  30. 30. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. Polynomial Expressions
  31. 31. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Polynomial Expressions
  32. 32. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Set x = (–3) in the expression, Polynomial Expressions
  33. 33. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3 Polynomial Expressions
  34. 34. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3 = 4(9) – 3(–27) Polynomial Expressions
  35. 35. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3 = 4(9) – 3(–27) = 36 + 81 = 117 Polynomial Expressions
  36. 36. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3 = 4(9) – 3(–27) = 36 + 81 = 117 Given a polynomial, each monomial is called a term. Polynomial Expressions
  37. 37. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3 = 4(9) – 3(–27) = 36 + 81 = 117 Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms Polynomial Expressions
  38. 38. Example B. Evaluate the polynomial 4x2 – 3x3 if x = –3. The polynomial 4x2 – 3x3 is the combination of two monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results. Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3 = 4(9) – 3(–27) = 36 + 81 = 117 Given a polynomial, each monomial is called a term. #xN ± #xN-1 ± … ± #x ± # terms Therefore the polynomial –3x2 – 4x + 7 has 3 terms, –3x2 , –4x and + 7. Polynomial Expressions
  39. 39. Each term is addressed by the variable part. Polynomial Expressions
  40. 40. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, Polynomial Expressions
  41. 41. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, Polynomial Expressions
  42. 42. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. Polynomial Expressions
  43. 43. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. Polynomial Expressions
  44. 44. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Polynomial Expressions
  45. 45. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Operations with Polynomials Polynomial Expressions
  46. 46. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Operations with Polynomials Polynomial Expressions
  47. 47. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. Operations with Polynomials Polynomial Expressions
  48. 48. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x Operations with Polynomials Polynomial Expressions
  49. 49. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Operations with Polynomials Polynomial Expressions
  50. 50. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. Operations with Polynomials Polynomial Expressions
  51. 51. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. So x + x2 stays as x + x2. Operations with Polynomials Polynomial Expressions
  52. 52. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. So x + x2 stays as x + x2. Note that we write 1xN as xN , –1xN as –xN. Operations with Polynomials Polynomial Expressions
  53. 53. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. So x + x2 stays as x + x2. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it with the coefficient. Operations with Polynomials Polynomial Expressions
  54. 54. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. So x + x2 stays as x + x2. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x Operations with Polynomials Polynomial Expressions
  55. 55. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. So x + x2 stays as x + x2. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x, Operations with Polynomials Polynomial Expressions
  56. 56. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. So x + x2 stays as x + x2. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x. Operations with Polynomials Polynomial Expressions
  57. 57. Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 . Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2. Unlike terms may not be combined. So x + x2 stays as x + x2. Note that we write 1xN as xN , –1xN as –xN. When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x. Operations with Polynomials When multiplying a number with a polynomial, we may expand using the distributive law: A(B ± C) = AB ± AC. Polynomial Expressions
  58. 58. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) Polynomial Expressions
  59. 59. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x Polynomial Expressions
  60. 60. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 Polynomial Expressions
  61. 61. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) Polynomial Expressions
  62. 62. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 Polynomial Expressions
  63. 63. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomial Expressions
  64. 64. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. Polynomial Expressions
  65. 65. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers Polynomial Expressions
  66. 66. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 Polynomial Expressions
  67. 67. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Polynomial Expressions
  68. 68. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Like–terms are terms where the variable parts are the same. Polynomial Expressions
  69. 69. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Like–terms are terms where the variable parts are the same. For example 3x2y3 + 5x2y3 = 8x2y3 Polynomial Expressions
  70. 70. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Like–terms are terms where the variable parts are the same. For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. Polynomial Expressions
  71. 71. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Like–terms are terms where the variable parts are the same. For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. We evaluate them by assigning numbers to x and/or y. Polynomial Expressions
  72. 72. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Like–terms are terms where the variable parts are the same. For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. We evaluate them by assigning numbers to x and/or y. If only one number is given, the result is a formula. Polynomial Expressions
  73. 73. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Like–terms are terms where the variable parts are the same. For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. We evaluate them by assigning numbers to x and/or y. If only one number is given, the result is a formula. If both numbers are given, then we get a numerical output. Polynomial Expressions
  74. 74. Example C. Expand and simplify. a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3 Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2. Like–terms are terms where the variable parts are the same. For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. We evaluate them by assigning numbers to x and/or y. If only one number is given, the result is a formula. If both numbers are given, then we get a numerical output. We may do this for x, y and z or even more variables. Polynomial Expressions
  75. 75. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 Polynomial Expressions
  76. 76. = 6xy – 8x2y + 2xy – 3xy2 Polynomial Expressions Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2
  77. 77. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 Polynomial Expressions
  78. 78. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Polynomial Expressions
  79. 79. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 Polynomial Expressions
  80. 80. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 Polynomial Expressions
  81. 81. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 Polynomial Expressions
  82. 82. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3 Polynomial Expressions
  83. 83. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3 We may put x = 2, y = 3 into the formula and do everything all over Polynomial Expressions
  84. 84. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3 We may put x = 2, y = 3 into the formula and do everything all over again or we may plug into y = 3 into part b which is easier. Polynomial Expressions
  85. 85. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3 We may put x = 2, y = 3 into the formula and do everything all over again or we may plug into y = 3 into part b which is easier. We will do the easy way. Polynomial Expressions
  86. 86. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3 We may put x = 2, y = 3 into the formula and do everything all over again or we may plug into y = 3 into part b which is easier. We will do the easy way. Input y = 3 into –16y – 6y2 Polynomial Expressions
  87. 87. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3 We may put x = 2, y = 3 into the formula and do everything all over again or we may plug into y = 3 into part b which is easier. We will do the easy way. –16(3) – 6(3)2 Input y = 3 into –16y – 6y2 we get Polynomial Expressions
  88. 88. Example D. Expand and simplify. a. 2(3xy – 4x2y) + 2xy – 3xy2 = 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2 b. Evaluate 8xy – 8x2y – 3xy2 if x = 2. Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2 = 16y – 32y – 6y2 = –16y – 6y2 c. Evaluate 8xy – 8x2y – 3xy2 if x = 2 and y = 3 We may put x = 2, y = 3 into the formula and do everything all over again or we may plug into y = 3 into part b which is easier. We will do the easy way. –16(3) – 6(3)2 Input y = 3 into –16y – 6y2 we get = –48 – 54 = –102 Polynomial Expressions
  89. 89. Ex. A. Evaluate each monomials with the given values. 3. 2x2 with x = 1 and x = –1 4. –2x2 with x = 1 and x = –1 5. 5y3 with y = 2 and y = –2 6. –5y3 with y = 2 and y = –2 1. 2x with x = 1 and x = –1 2. –2x with x = 1 and x = –1 7. 5z4 with z = 2 and z = –2 8. –5y4 with z = 2 and z = –2 B. Evaluate each monomials with the given values. 9. 2x2 – 3x + 2 with x = 1 and x = –1 10. –2x2 + 4x – 1 with x = 2 and x = –2 11. 3x2 – x – 2 with x = 3 and x = –3 12. –3x2 – x + 2 with x = 3 and x = –3 13. –2x3 – x2 + 4 with x = 2 and x = –2 14. –2x3 – 5x2 – 5 with x = 3 and x = –3 C. Expand and simplify. 15. 5(2x – 4) + 3(4 – 5x) 16. 5(2x – 4) – 3(4 – 5x) 17. –2(3x – 8) + 3(4 – 9x) 18. –2(3x – 8) – 3(4 – 9x) 19. 7(–2x – 7) – 3(4 – 3x) 20. –5(–2 – 8x) + 7(–2 – 11x) Polynomial Expressions
  90. 90. 21. x2 – 3x + 5 + 2(–x2 – 4x – 6) 22. x2 – 3x + 5 – 2(–x2 – 4x – 6) 23. 2(x2 – 3x + 5) + 5(–x2 – 4x – 6) 24. 2(x2 – 3x + 5) – 5(–x2 – 4x – 6) 25. –2(3x2 – 2x + 5) + 5(–4x2 – 4x – 3) 26. –2(3x2 – 2x + 5) – 5(–4x2 – 4x – 3) 27. 4(3x3 – 5x2) – 9(6x2 – 7x) – 5(– 8x – 2) 29. Simplify 2(3xy – xy2) – 2(2xy – xy2) then evaluated it with x = –1, afterwards evaluate it at (–1, 2) for (x, y) 30. Simplify x2 – 2(3xy – x2) – 2(y2 – xy) then evaluated it with y = –2, afterwards evaluate it at (–1, –2) for (x, y) 31. Simplify x2 – 2(3xy – z2) – 2(z2 – x2) then evaluated it with x = –1, y = – 2 and z = 3. Polynomial Expressions 28. –6(7x2 + 5x – 9) – 7(–3x2 – 2x – 7)

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