In these lessons, we will look at the Factor Theorem and how it relates to the Remainder Theorem. We will also show how to factor polynomials using the Factor Theorem.

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**What is the Factor Theorem?**

When f(*x*) is divided by *x* – *a*, we get
*a*) = 0 then the remainder is 0 and
*x* – *a* is a factor of f(*x*)

The Factor Theorem states that

**How to use the Factor Theorem and Remainder Theorem?**

The Factor Theorem and The Remainder Theorem

What the theorems are and how they can be used to find the linear factorization of a polynomial?

Factor F(x) = 2x^{3} − 3x^{2} − 5x + 6
This video demonstrates how to use the Factor Theorem to factor polynomials.

1) Factor P(x) = 3x^{3} − x^{2} − 19x + 8

1) Factor P(x) = 2x^{3} − 9x^{2} + x + 12

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)
Polynomials - Factor Theorem

Description and examples of the Factor Theorem

Prove that (x + 1) is a factor of P(x) = x^{2} + 2x + 1

Is (x + 2) a factor of x^{3} + 4x^{2} − x − 3?

The Factor Theorem

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.

Fully factor x^{4} − 3x^{3} − 7x^{2} + 15x + 18
Dividing Polynomials; Remainder and Factor Theorem.

You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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When f(

f(From the Remainder Theorem, we getx) = (x–a)Q(x) + remainder

f(If f(x) = (x–a)Q(x) + f(a)

f(We can then say thatx) = (x–a)Q(x)

The Factor Theorem states that

x–ais a factor of the polynomial f(x)if f(a) = 0

**Example:**

Determine whether *x* + 1 is a factor of the following polynomials.

a) 3*x*^{4} + *x*^{3} – *x*^{2} + 3*x* + 2

b) *x*^{6} + 2*x*(*x* – 1) – 4

**Solution:**

a) Let f(*x*) = 3*x*^{4} + *x*^{3} – *x*^{2} + 3*x* + 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*) = *x*^{6} + 2*x*(*x* – 1) – 4

g(–1) = (–1)6 + 2(–1)( –2) –4 = 1

Therefore, *x* + 1 is not a factor of g(*x*)

The Factor Theorem and The Remainder Theorem

What the theorems are and how they can be used to find the linear factorization of a polynomial?

Factor F(x) = 2x

1) Factor P(x) = 3x

1) Factor P(x) = 2x

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)

Description and examples of the Factor Theorem

Prove that (x + 1) is a factor of P(x) = x

Is (x + 2) a factor of x

The Factor Theorem

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.

Fully factor x

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