Factor Theorem: All You Need to Know

Factoring equations can be tricky and challenging, especially when your child needs to learn all the methods to solve them – and let’s face it, memorizing long formulas isn’t often their priority. 

But don’t worry! With the Factor Theorem, you’ll easily understand how best to approach these complex problems. In this blog post, we’ll discuss what exactly the Factor Theorem has to offer and why it is necessary for simplifying equations so that you and your child can gain a better understanding of this crucial mathematical concept. 

What is a Factor Theorem?

The factor theorem states that if a polynomial p(x) can be divided by (x-a) with no remainder, then p(a) = 0. Conversely, if p(a) = 0, then (x-a) is a factor of p(x). This theorem provides a way to find the roots, or x-values, that make p(x) equal to zero.

Proof of Factor Theorem:

The proof of the factor theorem involves showing that if p(x) is divisible by (x-a), then p(a) = 0, and conversely, if p(a) = 0, then (x-a) is a factor of p(x).

To show that if p(x) is divisible by (x-a), then p(a) = 0, consider the polynomial division:

p(x) = (x-a) q(x) + r, where q(x) and r are polynomials, and the degree of r is less than the degree of (x-a), which is 1.

Setting x = a in the equation above, we get:

p(a) = (a-a) q(a) + r = r.

Since the degree of r is less than the degree of (x-a), we have r = 0. Hence, p(a) = 0.

To show that if p(a) = 0, then (x-a) is a factor of p(x), consider the polynomial:

p(x) – p(a) = (x-a) q(x) for some polynomial q(x).

Since p(a) = 0, we have:

p(x) = (x-a) q(x).

This shows that (x-a) is a factor of p(x).

Therefore, the factor theorem is proven.

Example of Factor Theorem: Factorize y2 – 5y + 6

Let p(y) = y2 – 5y + 6. 

p(y) = (y – a)(y – b)

= y2 – by – ay + ab

On comparing the constants, we get ab = 6. 

Next, the factors of 6 are 1, 2, and 3. 

Now, p(2) = 22– (5 × 2) + 6 = 4 – 10 + 6 = 0. 

So, (y – 2) is a factor of p(y). 

Also, p(3) = 32 – (5 × 3) + 6 = 9 – 15 + 6 = 0.

 So, (y – 3) is also a factor of y2 – 5y + 6. 

Therefore, y2 – 5y + 6 = (y – 2)(y – 3)

Proof by Remainder Theorem:

We can also prove the factor theorem using the remainder theorem. The remainder theorem states that for a polynomial p(x) and a constant a, the remainder when p(x) is divided by (x-a) is equal to p(a).

Using this result, we can prove the factor theorem as follows:

If p(x) is divisible by (x-a), then the remainder when p(x) is divided by (x-a) is 0. Therefore, p(a) = 0.

Conversely, if p(a) = 0, then the remainder when p(x) is divided by (x-a) is 0. This means that (x-a) is a factor of p(x).

Therefore, the factor theorem is proven using the remainder theorem.

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How to Use Factor Theorem 

Write the polynomial: Write the polynomial that you want to find the roots of or the polynomial that you want to factor.

Choose a value of “a”: Choose a constant “a” to test as a possible root of the polynomial.

Evaluate p(a): Substitute the value of “a” into the polynomial and evaluate the expression to get p(a).

Check if p(a) = 0: If p(a) = 0, then (x-a) is a factor of the polynomial.

Factor the polynomial: If (x-a) is a factor of the polynomial, you can use polynomial division to factor the polynomial. Divide the polynomial by (x-a) and write the result in the form p(x) = (x-a) q(x) + r, where q(x) is the quotient and r is the remainder. If r = 0, then (x-a) is a factor of the polynomial with no remainder.

Repeat steps 2-5: Repeat steps 2-5 with different values of “a” until you have found all the roots of the polynomial.

Construct the factors: Use the roots that you found in step 6 to construct the factors of the polynomial. The polynomial can be written as the product of the factors.

In summary, the factor theorem provides a way to find the roots of a polynomial and to factor the polynomial. The process involves evaluating the polynomial at a constant “a,” checking if the result equals zero, and if so, using polynomial division to factor the polynomial.

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Why Do We Use Factor Theorem

The factor theorem is used for several reasons:

Finding roots of polynomials: The factor theorem provides a method for finding the roots, or x-values, that make a polynomial equal to zero.

Factoring polynomials: The factor theorem can be used to factor polynomials by finding the roots of the polynomial and then constructing the corresponding factors.

Simplifying polynomials: By factoring polynomials using the factor theorem, it is possible to simplify the expression and make it easier to work with.

Solving polynomial equations: The factor theorem can be used to solve polynomial equations by factoring the polynomial on one side of the equation and then using the roots to find the solution.

In summary, a factor theorem is a useful tool in algebra for finding the roots of polynomials, factoring polynomials, simplifying expressions, and solving polynomial equations.

What are the other methods to find the factors of polynomials?

1. Synthetic Division: This is a method for dividing a polynomial by a linear factor, which can be used to find the factors of the polynomial.
2. Long Division: This is a method for dividing a polynomial by another polynomial, which can be used to find the factors of the polynomial.
3. Rational Root Theorem: This theorem states that if a polynomial has a rational root, then the root must divide the leading coefficient and the constant term of the polynomial. This theorem can be used to narrow down the search for roots.
4. Descartes’ Rule of Signs: This rule states that the number of positive roots of a polynomial is less than or equal to the number of sign changes in the coefficients of the polynomial.
5. Factoring by Grouping: This is a method for factoring a polynomial by grouping terms together and then using the distributive property to factor the terms.
6. Special Factoring Techniques: There are several special factoring techniques, such as factoring by substitution, factoring by grouping, and factoring using the sum and difference of cubes.

In conclusion, the Factor Theorem is a crucial mathematical concept that helps find the roots of polynomials and factor them. 

The Factor Theorem can be proven using polynomial division or the remainder theorem. The factor theorem is widely used because of its effectiveness in solving complex polynomial problems.

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