Factor Theorem

Remainder Theorem And Factor Theorem

What Is The Factor Theorem?

When f(x) is divided by (x – a), we get
f(x) = (x – a)Q(x) + remainder

From the Remainder Theorem we get
f(x) = (x – a)Q(x) + f(a)

If f(a) = 0 then the remainder is 0 and
f(x) = (x – a)Q(x)

We can then say that (x – a) is a factor of f(x)

The Factor Theorem states that
(x – a) is a factor of the polynomial f(x) if and only if f(a) = 0

Take note that the following statements are equivalent for any polynomial f(x).

• (x – a) is a factor of f(x).
• The remainder is zero when f(x) is divided by (x – a).
• f(a) = 0.
• The solution to f(x) = 0 is a.
• The zero of the function f(x) is a.

Example:
Determine whether x + 1 is a factor of the following polynomials.
a) 3x⁴ + x³ – x² + 3x + 2
b) x⁶ + 2x(x – 1) – 4

Solution:
a) Let f(x) = 3x⁴ + x³ – x² + 3x + 2
f(–1) = 3(–1)4 + (–1)3 – (–1)2 +3(–1) + 2
= 3(1) + (–1) – 1 – 3 + 2 = 0
Therefore, x + 1 is a factor of f(x)

b) Let g(x) = x⁶ + 2x(x – 1) – 4
g(–1) = (–1)6 + 2(–1)( –2) –4 = 1
Therefore, x + 1 is not a factor of g(x)

Videos

How To Use The Factor Theorem And Remainder Theorem?

What the theorems are and how they can be used to find the linear factorization of a polynomial?
The Remainder Theorem states that if a polynomial, f(x), is divided by x - k, the remainder is equal to f(k).
The Factor Theorem states that the polynomial x - k is a factor of the polynomial f(x) if and only if f(k) = 0.

Example:
Let f(x) = 2x³ − 3x² − 5x + 6
Is x - 1 a factor?
Find all the other factors.

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How To Use The Factor Theorem To Factor Polynomials?

Examples:

1. Factor P(x) = 3x³ − x² − 19x + 8
   
2. Factor P(x) = 2x³ − 9x² + x + 12
   

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How To Find Remaining Factors Of A Polynomial?

A lesson on the factor theorem and completely factoring a polynomial.

1. To learn the connection between the factor theorem and the remainder theorem.
2. To learn how to use the factor theorem to determine if a binomial is a factor of a given polynomial or not.
3. To use synthetic division, along with the factor theorem to help factor a polynomial.

Example:
Fully factor x⁴ − 3x³ − 7x² + 15x + 18

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Application Of The Factor Theorem

How to use the factor theorem to determine if x - c is a factor of the polynomial f?
Examples:

1. f(x) = 4x³ - 3x² - 8x + 4, c = 3
2. f(x) = 3x⁴ - 6x³ - 5x + 10, c = 1
3. f(x) = 3x⁶ + 2x³ - 176, c = -2
4. f(x) = 4x⁶ - 64x⁴ - x² - 16, c = 4
5. f(x) = 2x⁴ - x³ - 2x - 1, c = -1/2

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How To Explain The Factor Theorem?

If f(x) is a polynomial and f(p) = 0 then (x − p) is a factor of f(x)
If f(x) is a polynomial and f(−q) = 0 then (x + q) is a factor of f(x)

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Description And Examples Of The Factor Theorem

Examples:
Prove that (x + 1) is a factor of P(x) = x² + 2x + 1
Is (x + 2) a factor of x³ + 4x² − x − 3?

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