# Factor Theorem

In mathematics, the factor theorem helps in fully factoring a polynomial. This theorem links the polynomial's factors to its zeros.

## Introduction to Factor Theorem

The Factor Theorem states that for a polynomial *f(x)* of degree *n ≥ 1* and any real number *a*, *(x - a)* is a factor of *f(x)* if and only if *f(a) = 0*. This means that if *(x - a)* is a factor of *f(x)*, then substituting *a* into the polynomial results in zero, confirming the theorem.

Let's explore this theorem with some examples to understand its application.

## What is the Factor Theorem?

The Factor Theorem is a method used to determine the factors and roots of polynomials. It states that a polynomial *f(x)* has a factor *(x - a)* if and only if *f(a) = 0*. This is a key concept in polynomial factorization.

### Proof

We will now prove the Factor Theorem, which allows us to factorize polynomials. Suppose a polynomial *f(x)* is divisible by *(x - c)*, then *f(c) = 0*. According to the Remainder Theorem:

*f(x) = (x - c)q(x) + f(c)*

Here, *f(x)* is the polynomial, and *q(x)* is the quotient polynomial. Since *f(c) = 0*:

*f(x) = (x - c)q(x) + 0*

*f(x) = (x - c)q(x)*

Thus, *(x - c)* is a factor of the polynomial *f(x)*.

### Another Method

Using the Remainder Theorem again:

*f(x) = (x - c)q(x) + f(c)*

If *(x - c)* is a factor of *f(x)*, then the remainder must be zero, meaning *(x - c)* exactly divides *f(x)*. Therefore, *f(c) = 0*.

The following statements are equivalent for any polynomial *f(x)*:

- When
*f(x)*is divided by*(x - c)*, the remainder is zero. *(x - c)*is a factor of*f(x)*.*c*is a root or solution of*f(x)*.*c*is a zero of the function*f(x)*or*f(c) = 0*.

## How to Use Factor Theorem

To find the factors of a polynomial using the factor theorem, follow these steps:

- If
*f(-c) = 0*, then*(x + c)*is a factor of*f(x)*. - If
*pdc = 0*, then*(cx - d)*is a factor of*f(x)*. - If
*p - dc = 0*, then*(cx + d)*is a factor of*f(x)*. - If
*p(c) = 0*and*p(d) = 0*, then*(x - c)*and*(x - d)*are factors of*p(x)*.

Instead of using polynomial long division, the factor theorem and synthetic division are often easier ways to find factors. The factor theorem helps identify known zeros of a polynomial, reducing its degree and making it simpler to solve.

When dividing a polynomial by a binomial factor, if the remainder is zero, the binomial is indeed a factor of the polynomial. This is known as the factorization theorem.

## Other Methods to Find Factors

Besides the factor theorem, you can use:

**Polynomial Long Division****Synthetic Division**

### Polynomial Long Division Example

For *f(x) = x ^{2} + 2x - 15*:

Solve *x ^{2} + 2x - 15 = 0*:

*x ^{2} + 5x - 3x - 15 = 0*

*x(x + 5) - 3(x + 5) = 0*

*(x - 3)(x + 5) = 0*

Thus, *x = -5* or *x = 3* are the roots.

### Synthetic Division Example

Using *f(x) = x ^{2} + 2x - 15* and dividing by

*x - 3*:

Place 3 on the left and use the coefficients 1, 2, and -15 from *f(x)*.

Perform synthetic division to verify that the remainder is zero, confirming that *x - 3* is a factor.

Because the remainder is zero, 3 is a solution of the polynomial. Solving equations of degree 3 or higher is more complex, so we often use simpler linear and quadratic equations instead.

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## Frequently Asked Questions on Factor Theorem

The factor theorem states that a polynomial f(x) has (x - a) as a factor if and only if f(a) = 0. This means that if a number 'a' makes the polynomial equal to 0, then (x - a) is a factor of the polynomial.

For example, if f(x) = x^{3} - 2x^{2}

The zero factor theorem states that if the product of two factors is equal to zero, then at least one of the factors must be equal to zero. This is a fundamental principle used in the factor theorem.

A factor rule is a mathematical principle that describes how to find the factors of a polynomial expression. The factor theorem is one such rule that relates the factors of a polynomial to its zeros or roots.

The main rules of the factor theorem are:

- If f(a) = 0, then (x - a) is a factor of f(x).
- If (x - a) is a factor of f(x), then f(a) = 0.
- The remainder when f(x) is divided by (x - a) is equal to f(a).