The document defines key polynomial vocabulary including: - Terms are numbers or products of numbers and variables raised to powers. Coefficients are numerical factors of terms. Constants are terms that are only numbers. - Polynomials are sums of terms involving variables raised to whole number exponents, with no variables in denominators. - Types of polynomials include monomials (1 term), binomials (2 terms), and trinomials (3 terms). Degree is the largest exponent of any term. - Operations on polynomials include adding/subtracting like terms, multiplying using distribution and FOIL, dividing using long division, and special products like (a+b)2 and (a+b)(a

1. Let’s Define it.

2. P O LY N O M I A L V O C A B U L A R Y
Term – a number or a product of a number and variables
raised to powers
Coefficient – numerical factor of a term
Constant – term which is only a number
Polynomial is a sum of terms involving variables raised to
a whole number exponent, with no variables appearing
in any denominator.

3. P O LY N O M I A L V O C A B U L A R Y
In the polynomial 7x5 + x2y2 – 4xy + 7
There are 4 terms: 7x5, x2y2, -4xy and 7.
The coefficient of term 7x5 is 7,
of term x2y2 is 1,
of term –4xy is –4 and
of term 7 is 7.
7 is a constant term.

4. T Y P E S O F P O LY N O M I A L S
Monomial is a polynomial with 1 term.
Binomial is a polynomial with 2 terms.
Trinomial is a polynomial with 3 terms.
Multinomial is a polynomial with 4 or more terms.

5. DEGREES
Degree of a term
To find the degree, take the sum of the exponents on the
variables contained in the term.
Degree of a constant is 0.
Degree of the term 5a4b3c is 8 (remember that c can be
written as c1).
Degree of a polynomial
To find the degree, take the largest degree of any term of the
polynomial.
Degree of 9x3 – 4x2 + 7 is 3.

6. E VA L U AT I N G P O LY N O M I A L S
Evaluating a polynomial for a particular value involves
replacing the value for the variable(s) involved.
Example
Find the value of 2x3 – 3x + 4 when x = 2.
2x3 – 3x + 4 = 2( 2)3 – 3( 2) + 4
= 2( 8) + 6 + 4
= 6

7. COMBINING LIKE TERMS
Like terms are terms that contain exactly the same variables raised
to exactly the same powers.
Warning!
Only like terms can be combined through addition and
subtraction.
Example
Combine like terms to simplify.
x2y + xy – y + 10x2y – 2y + xy
= x2y + 10x2y + xy + xy – y – 2y (Like terms are grouped together)
= (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y = 11x2y + 2xy – 3y

8. ADDING AND SUBTRACTING
POLYNOMIALS
Let’s Add and Subtract!

9. ADDING AND SUBTRACTING POLYNOMIALS
Adding Polynomials
Combine all the like terms.
Subtracting Polynomials
Change the signs of the terms of the
polynomial being subtracted, and then
combine all the like terms.

10. ADDING AND SUBTRACTING POLYNOMIALS
Example
Add or subtract each of the following, as indicated.
1) (3x – 8) + (4x2 – 3x +3) = 3x – 8 + 4x2 – 3x + 3
= 4x2 + 3x – 3x – 8 + 3
= 4x2 – 5
2) 4 – (– y – 4) = 4 + y + 4 = y + 4 + 4 = y + 8
3) (– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7)
= – a2 + 1 – a2 + 3 + 5a2 – 6a + 7
= – a2 – a2 + 5a2 – 6a + 1 + 3 + 7 = 3a2 – 6a + 11

11. ADDING AND SUBTRACTING POLYNOMIALS
In the previous examples, after discarding the
parentheses, we would rearrange the terms so
that like terms were next to each other in the
expression.
You can also use a vertical format in arranging
your problem, so that like terms are aligned with
each other vertically.

12. MULTIPLYING POLYNOMIALS
Let’s Multiply!

13. M U LT I P LY I N G P O LY N O M I A L S
Multiplying polynomials
• If all of the polynomials are monomials, use
the associative and commutative properties.
• If any of the polynomials are not monomials,
use the distributive property before the
associative and commutative properties.
Then combine like terms.

14. Multiplying Polynomials
Example
Multiply each of the following.
1) (3x2)(– 2x) = (3)(– 2)(x2 · x) = – 6x3
2) (4x2)(3x2 – 2x + 5)
= (4x2)(3x2) – (4x2)(2x) + (4x2)(5) (Distributive property)
= 12x4 – 8x3 + 20x2 (Multiply the monomials)
3) (2x – 4)(7x + 5) = 2x(7x + 5) – 4(7x + 5)
= 14x2 + 10x – 28x – 20
= 14x2 – 18x – 20

15. Multiplying Polynomials
Example
Multiply (3x + 4)2
Remember that a2 = a · a, so (3x + 4)2 = (3x + 4)(3x + 4).
(3x + 4)2 = (3x + 4)(3x + 4) = (3x)(3x + 4) + 4(3x + 4)
= 9x2 + 12x + 12x + 16
= 9x2 + 24x + 16

16. Multiplying Polynomials
Example
Multiply (a + 2)(a3 – 3a2 + 7).
(a + 2)(a3 – 3a2 + 7) = a(a3 – 3a2 + 7) + 2(a3 – 3a2 + 7)
= a4 – 3a3 + 7a + 2a3 – 6a2 +
14a4 – a3 – 6a2 + 7a + 14
=

17. Multiplying Polynomials
Example
Multiply (3x – 7y)(7x + 2y)
(3x – 7y)(7x + 2y) = (3x)(7x + 2y) – 7y(7x + 2y)
= 21x2 + 6xy – 49xy + 14y2
= 21x2 – 43xy + 14y2

18. Multiplying Polynomials
Example
Multiply (5x – 2z)2
(5x – 2z)2 = (5x – 2z)(5x – 2z) = (5x)(5x – 2z) – 2z(5x – 2z)
= 25x2 – 10xz – 10xz + 4z2
= 25x2 – 20xz + 4z2

19. Multiplying Polynomials
Example
Multiply (2x2 + x – 1)(x2 + 3x + 4)
(2x2 + x – 1)(x2 + 3x + 4)
= (2x2)(x2 + 3x + 4) + x(x2 + 3x + 4) – 1(x2 + 3x + 4)
= 2x4 + 6x3 + 8x2 + x3 + 3x2 + 4x – x2 – 3x – 4
= 2x4 + 7x3 + 10x2 + x – 4

20. SPECIAL PRODUCTS
Let’s multiply!

21. THE FOIL METHOD
When multiplying 2 binomials, the distributive
property can be easily remembered as the FOIL
method.
F – product of First terms
O – product of Outside terms
I – product of Inside terms
L – product of Last terms

22. Using the FOIL Method
Example
Multiply (y – 12)(y + 4)
(y – 12)(y + 4) Product of First terms is y2
(y – 12)(y + 4) Product of Outside terms is 4y
(y – 12)(y + 4) Product of Inside terms is -12y
(y – 12)(y + 4) Product of Last terms is -48
F O I L
(y – 12)(y + 4) = y2 + 4y – 12y – 48
= y2 – 8y – 48

23. Using the FOIL Method
Example
Multiply (2x – 4)(7x + 5)
F L F O I L
(2x – 4)(7x + 5) = 2x(7x) + 2x(5) – 4(7x) – 4(5)
I
O = 14x2 + 10x – 28x – 20
= 14x2 – 18x – 20
We multiplied these same two binomials together in the
previous section, using a different technique, but arrived at the
same product.

24. Special Products
In the process of using the FOIL method on products of
certain types of binomials, we see specific patterns that
lead to special products.
Squaring a Binomial
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
Multiplying the Sum and Difference of Two Terms
(a + b)(a – b) = a2 – b2

25. Special Products
Although you will arrive at the same
results for the special products by using
the techniques of this section or last
section, memorizing these products can
save you some time in multiplying
polynomials.

26. DIVIDING POLYNOMIALS
Let’s divide!

27. D I V I D I N G P O LY N O M I A L S
Dividing a polynomial by a monomial
Divide each term of the polynomial separately by the
monomial.
3 3
12 a 36 a 15 12 a 36 a 15
Example
3a 3a 3a 3a
2 5
4a 12
a

28. DIVIDING POLYNOMIALS
D I V I D I N G P O LY N O M I A L S
Dividing a polynomial by a polynomial
other than a monomial uses a “long
division” technique that is similar to the
process known as long division in
dividing two numbers, which is reviewed
on the next slide.

29. DIVIDING POLYNOMIALS
D I V I D I N G P O LY N O M I A L S
168 Divide 43 into 72.
Multiply 1 times 43.
43 7256 Subtract 43 from 72.
43 Bring down 5.
29 5 Divide 43 into 295.
258 Multiply 6 times 43.
Subtract 258 from 295.
37 6 Bring down 6.
344 Divide 43 into 376.
32 Multiply 8 times 43.
Subtract 344 from 376. We then write our result as
32
Nothing to bring down. 168 .
43

30. Dividing P O L Y N O M I A L S
DIVIDING
Polynomials
As you can see from the previous example, there is
a pattern in the long division technique.
Divide
Multiply
Subtract
Bring down
Then repeat these steps until you can’t bring
down or divide any longer.
We will incorporate this same repeated technique
with dividing polynomials.

31. DIVIDING POLYNOMIALS
D I V I D I N G P O LY N O M I A L S
4x 5 Divide 7x into 28x2.
Multiply 4x times 7x+3.
2
7x 3 28 x 23 x 15 Subtract 28x2 + 12x from 28x2 – 23x.
2 Bring down – 15.
28 x 12 x Divide 7x into –35x.
35 x 15 Multiply – 5 times 7x+3.
Subtract –35x–15 from –35x–15.
35 x 15 Nothing to bring down.
So our answer is 4x – 5.

32. Dividing P O L Y N O M I A L S
DIVIDING
Polynomials
2 x 10 Divide 2x into 4x2.
2 Multiply 2x times 2x+7.
2 x 7 4x 6x 8 Subtract 4x2 + 14x from 4x2 – 6x.
2
4 x 14 x Bring down 8.
20 x 8 Divide 2x into –20x.
20 x 70 Multiply -10 times 2x+7.
Subtract –20x–70 from –20x+8.
78 Nothing to bring down.
We write our final answer as 2 x 10 78
( 2 x 7)

33. THE END
GOODBYE! 