Factor Theorem - Algebra I | Grade 9 | Luna Learn 24

Factor Theorem

Grade 9 Algebra I, USA High Schools

1. Description

The Factor Theorem is a powerful tool in algebra that connects the factors of a polynomial with its roots (or zeros). It helps us determine whether a given binomial is a factor of a polynomial. If substituting a specific value for \( x \) into the polynomial yields zero, then that value is a root, and the corresponding binomial is a factor of the polynomial.

2. Outline

Definition of Factor Theorem
Using the Factor Theorem to determine factors
Examples and exercises
Homework and revision
Video explanation

3. Definition of Factor Theorem

The Factor Theorem states that a polynomial \( f(x) \) has a factor \( (x - c) \) if and only if \( f(c) = 0 \). In other words, \( c \) is a root of the polynomial if and only if \( (x - c) \) is a factor of the polynomial.

4. Examples

Example 1: Determine if \( x - 2 \) is a factor of \( f(x) = x^3 - 4x^2 + x + 6 \).

Solution: Evaluate \( f(2) \):

\( f(2) = 2^3 - 4(2)^2 + 2 + 6 = 8 - 16 + 2 + 6 = 0 \)

Since \( f(2) = 0 \), \( x - 2 \) is a factor of the polynomial.

Example 2: Is \( x + 3 \) a factor of \( f(x) = x^2 + 5x + 6 \)?

Solution: Evaluate \( f(-3) \):

\( f(-3) = (-3)^2 + 5(-3) + 6 = 9 - 15 + 6 = 0 \)

Since \( f(-3) = 0 \), \( x + 3 \) is a factor of the polynomial.

Example 3: Determine if \( x - 1 \) is a factor of \( f(x) = x^3 - 7x + 6 \).

Solution: Evaluate \( f(1) \):

\( f(1) = 1^3 - 7(1) + 6 = 1 - 7 + 6 = 0 \)

Since \( f(1) = 0 \), \( x - 1 \) is a factor of the polynomial.

5. Exercises

Exercise 1: Determine if \( x - 4 \) is a factor of \( f(x) = x^3 - 12x + 16 \).

Solution: Evaluate \( f(4) \):

\( f(4) = 4^3 - 12(4) + 16 = 64 - 48 + 16 = 32 \)

Since \( f(4) \neq 0 \), \( x - 4 \) is not a factor of the polynomial.

Exercise 2: Is \( x + 5 \) a factor of \( f(x) = x^2 - 25 \)?

Solution: Evaluate \( f(-5) \):

\( f(-5) = (-5)^2 - 25 = 25 - 25 = 0 \)

Since \( f(-5) = 0 \), \( x + 5 \) is a factor of the polynomial.

Exercise 3: Determine if \( x - 6 \) is a factor of \( f(x) = x^2 - 36 \).

Solution: Evaluate \( f(6) \):

\( f(6) = 6^2 - 36 = 36 - 36 = 0 \)

Since \( f(6) = 0 \), \( x - 6 \) is a factor of the polynomial.

6. Homework

Solve the following problems and submit your answers in the comments section below:

Determine if \( x - 3 \) is a factor of \( f(x) = x^3 - 9x + 27 \).
Is \( x + 2 \) a factor of \( f(x) = x^2 + 4x + 4 \)?
Determine if \( x - 5 \) is a factor of \( f(x) = x^3 - 15x + 25 \).

7. Revision

Review the Factor Theorem and practice applying it to different polynomials. Make sure you understand how to determine if a binomial is a factor of a polynomial using substitution.