10.6 Introduction to Factoring Polynomials

Factoring Polynomials

Factoring is the reverse of multiplying. Instead of expanding expressions, you're breaking a polynomial down into simpler pieces that multiply together to give the original. This skill shows up constantly in algebra and beyond, so building a solid foundation now pays off.

Greatest Common Factor (GCF) Identification

The greatest common factor is the largest factor that divides evenly into every term of a polynomial. Finding the GCF is always your first step when factoring any expression.

Here's how to find it:

1. List the factors of each coefficient (the number part). Pick the largest number that appears in every list.
2. Look at the variables. For each variable that appears in all terms, take the one with the lowest exponent.
3. Multiply the results from steps 1 and 2 together. That's your GCF.

For example, in $6x^3 + 9x^2$, the coefficients are 6 and 9. The largest number dividing both is 3. The variable $x$ appears in both terms, and the lowest exponent is 2. So the GCF is $3x^2$.

Factoring Out Common Factors

Once you've found the GCF, you divide each term by it and write the polynomial as a product.

1. Find the GCF of all terms.
2. Divide each term by the GCF.
3. Write the result as: GCF × (what's left).

Example: Factor $10x^3 + 15x^2$.

• The GCF of 10 and 15 is 5. Both terms have at least $x^2$. So the GCF is $5x^2$.
• Divide each term: $10x^3 \div 5x^2 = 2x$ and $15x^2 \div 5x^2 = 3$.
• Factored form: $5x^2(2x + 3)$.

You can always check your answer by distributing the GCF back through the parentheses. If you get the original expression, you factored correctly.

Techniques for Polynomial Factoring

Beyond pulling out a GCF, there are a few other methods you'll use depending on what the polynomial looks like.

Factoring trinomials (three terms): For a trinomial like $x^2 + bx + c$, you're looking for two numbers that multiply to $c$ and add to $b$.

• Example: $x^2 + 5x + 6$. You need two numbers that multiply to 6 and add to 5. That's 2 and 3, so: $x^2 + 5x + 6 = (x + 2)(x + 3)$.

Factoring by grouping (four terms): When you have four terms, split them into two pairs, factor each pair separately, then look for a common binomial factor.

1. Group: $(ax + ay) + (bx + by)$
2. Factor each group: $a(x + y) + b(x + y)$
3. Factor out the common binomial: $(a + b)(x + y)$

Example: Factor $2x^3 + 3x^2 - 8x - 12$.

1. Group: $(2x^3 + 3x^2) + (-8x - 12)$

2. Factor each group: $x^2(2x + 3) - 4(2x + 3)$

3. Factor out $(2x + 3)$: $(x^2 - 4)(2x + 3)$

Notice that $x^2 - 4$ is a difference of squares, so it factors further into $(x - 2)(x + 2)$. The fully factored form is $(x - 2)(x + 2)(2x + 3)$.

Advanced Factoring Methods and Related Concepts

These topics go beyond the basics but are worth knowing as you move forward in algebra.

• Factor theorem: If plugging a value $r$ into a polynomial gives you zero, then $(x - r)$ is a factor. This connects roots and factors directly.
• Rational root theorem: Gives you a list of possible rational roots to test, which helps when factoring higher-degree polynomials.
• Synthetic division: A shortcut for dividing a polynomial by a linear factor like $(x - r)$. It's faster than long division once you get the hang of it.
• Quadratic formula: When a quadratic won't factor neatly, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ finds the roots directly. The expression under the square root, $b^2 - 4ac$, is called the discriminant and tells you whether the roots are real or not.