2. Let us identify whether the following
expressions are polynomials or not.
EXPRESSION POLYNOMIAL OR NOT REASON
x2 + 2x +5
5x + 1
(x7 + 2x4 – 5) (3x)
= 5x – 2 +1
= 3x½ +2
(6x2 +3x) ÷ (3x)
POLYNOMIAL
POLYNOMIAL
POLYNOMIAL
POLYNOMIAL
All exponents are positive integers
All exponents are positive integers
All exponents are positive integers
Can be reduced to an expression whose
exponents are positive integers
Presence of negative exponent
Presence of fractional exponent
8 POLYNOMIAL
NOT A POLYNOMIAL
NOT A POLYNOMIAL
8 = 8x0 ;exponent zero is a whole
number!
𝟓
𝒙𝟐
+ 𝟏
𝟑 𝒙 + 𝟐
3. So what is a polynomial?
A polynomial is an algebraic expression that
contains a specific number of terms each of
which is of the form axn, where a is a real
number and n is a whole number.
4. Name the degree, the leading coefficient, and the
constant term of each of the following polynomials.
POLYNOMIAL DEGREE OF
POLYNOMIAL
LEADING COEFFICIENT CONSTANT
x3 – 9x2 + 2x + 17
–3x – 5
11x7
m2 + 9m – 3
99
3
1
7
2
0
1
– 3
11
1
99 99
– 3
0
– 5
17
5. REMEMBER!
The degree of the polynomial is the highest degree among degrees of the
terms of the polynomial.
The leading coefficient is the numerical coefficient of the term with the
highest degree, and the constant term is the numerical coefficient of the
term whose exponent is zero.
The constant term is the numerical coefficient of the term whose exponent is
zero.
6. OPERATIONS ON POLYNOMIALS: SYNTHETIC DIVISION
𝟏. )
𝟒𝒙𝟐 + 𝟐𝒙𝟒 − 𝟏
𝒙 − 𝟏
Arrange the terms in descending powers, providing
zeroes for missing terms.
𝟐𝒙𝟒 + 𝟎𝒙𝟑 + 𝟒𝒙𝟐 + 𝟎𝒙 − 𝟏
𝒙 − 𝟏
Copy the numerical coefficients of the
dividend/numerator.
Find the root associated with the divisor. This serves as
the divisor of the Synthetic Division process.
Perform the Synthetic Division.
The coefficients of the quotient are
the values in the bottom row.
degree of quotient = degree of dividend – degree of divisor
13. Is 34 a factor of 170?
SOLUTION: 170 ÷ 34 = 5
So, 34 is a factor of 170.
14. Is 14 a factor of 156?
SOLUTION: 156 ÷ 14 = 11, remainder 2
So, 14 is NOT a factor of 156.
15. So, 12 is a factor of 96.
Is 12 a factor of 96?
SOLUTION: 96 ÷ 12 = 8
16. SOLUTION: 485 ÷ 27 = 17, remainder 26
So, 27 is NOT a factor of 485.
Is 27 a factor of 485?
17. How can we say that a number is a factor
of another number?
One number is a factor of another number if
there is no remainder after dividing the two
numbers.
18. How can we say that a binomial is a factor
of a polynomial?
The binomial is a factor of a polynomial if there
no remainder after dividing the polynomial by
the binomial.
19. To determine whether a binomial is a
factor of the polynomial, we use the
Factor Theorem.
To determine the remainder after dividing
the polynomial by a binomial, we use the
Remainder Theorem.
20. The Remainder Theorem
If a polynomial p(x) is divided by the binomial
x – c, then the remainder is equal to p(c).
The Factor Theorem
If p(c) = 0, then x – c is a factor of p(x).
Remember that p(c) = remainder!
21. EXERCISES:
Find the remainder when the polynomial is
divided by the given binomial, then
determine whether the given binomial is a
factor of a polynomial or not.
22. p(x) = x3 – 6x2 + 11x – 6
Remainder:
Factor or Not:
x + 2 = 0
x = –2
1.) p(x) = x3 – 6x2 + 11x – 6; x + 2
p(–2) = (–2)3 – 6(–2)2 + 11(–2) – 6
p(–2) = –60
x + 2 is NOT a factor of p(x)
→ REMAINDER
23. 2.) p(x) = x3 – 4x2 – 2x + 5; x – 1
Remainder:
Factor or Not:
x – 1 = 0
x = 1
p(x) = x3 – 4x2 – 2x + 5
p(1) = (1)3 – 4(1)2 – 2(1) + 5
p(1) = 0
x – 1 is a factor of p(x)
→ REMAINDER
24. 3.) p(x) = 2x3 – 7x2 + 3x + 4; x – 1
Remainder:
Factor or Not:
x – 1 = 0
x = 1
p(x) = 2x3 – 7x2 + 3x + 4
p(1) = 2(1)3 – 7(1)2 + 3(1) + 4
p(1) = 2
x – 1 is NOT a factor of p(x)
→ REMAINDER
25. 4.) p(x) = x4 + 2x3 – 8x – 16; x + 2
Remainder:
Factor or Not:
x + 2 = 0
x = –2
p(x) = x4 + 2x3 – 8x – 16
p(–2) = (–2)4 + 2(–2)3 – 8(–2) – 16
p(–2) = 0
x + 2 is a factor of p(x)
→ REMAINDER
26. 5.) p(x) = x3 – x2 – 5x – 3; x + 3
Remainder:
Factor or Not:
x + 3 = 0
x = –3
p(x) = x3 – x2 – 5x – 3
p(–3) = (–3)3 – (–3)2 – 5(–3) – 3
p(–3) = –24
x + 3 is NOT a factor of p(x)
→ REMAINDER
27. p(x) = 2x2 + x – 5
Remainder:
Factor or Not:
x = 4
x – 4 = 0
6.) p(x) = 2x2 + x – 5; x – 4
p(4) = 2(4)2 + 4 – 5
p(4) = 31
x – 4 is NOT a factor of p(x)
→ REMAINDER
28. 7.) p(x) = 3x4 + 9x3 – 9x + 9; x – 2
Remainder:
Factor or Not: x – 2 is NOT a factor of p(x)
x – 2 = 0
x = 2
p(x) = 3x4 + 9x3 – 9x + 9
p(2) = 3(2)4 + 9(2)3 – 9(2) + 9
p(2) = 32 → REMAINDER
29. x – 2 = 0
Remainder:
Factor or Not: x – 2 is a factor of p(x)
8.) p(x) = 2x3 + 5x2 – 6x – 24 ; x – 2
x = 2
p(x) = 2x3 + 5x2 – 6x – 24
p(2) = 2(2)3 + 5(2)2 – 6(2) – 24
p(2) = 0 → REMAINDER
