Complete Guide to Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler factors that, when multiplied together, give you the original polynomial. It’s a key skill in algebra that makes solving equations easier.

1. Greatest Common Factor (GCF)

The first step in factoring any polynomial is to check if there is a common factor that can be factored out of all terms. This is called the Greatest Common Factor (GCF).

Steps to factor out the GCF:

1. Find the GCF of the coefficients.
2. Look for the smallest power of each variable that appears in all terms.
3. Factor out the GCF.

Example:

Factor out the GCF in the expression $6x^3 + 9x^2$.

1. The GCF of $6$ and $9$ is $3$.
2. The smallest power of $x$ common to both terms is $x^2$.

So, we factor out $3x^2$:

2. Factoring Trinomials

A trinomial is a polynomial with three terms. The most common form of a trinomial is:

2.1 Factoring Trinomials when $a = 1$:

When $a = 1$, you’re looking for two numbers that:

1. Multiply to give $c$.
2. Add to give $b$.

Example:

Factor $x^2 + 7x + 12$.

• Look for two numbers that multiply to $12$ and add to $7$. These numbers are $3$ and $4$.
• Rewrite the trinomial as:

2.2 Factoring Trinomials when $a \neq 1$:

When $a \neq 1$, you’ll need to use factoring by grouping. Follow these steps:

1. Multiply $a$ and $c$.
2. Find two numbers that multiply to $a \cdot c$ and add to $b$.
3. Rewrite the middle term using those numbers.
4. Factor by grouping.

Example:

Factor $2x^2 + 7x + 3$.

1. Multiply $2 \times 3 = 6$.
2. Find two numbers that multiply to $6$ and add to $7$. These numbers are $6$ and $1$.
3. Rewrite the middle term:

4. Factor by grouping:

3. Factoring Difference of Squares

A difference of squares looks like this:

It factors into:

Example:

Factor $x^2 - 16$.

Since $16$ is a perfect square, we can write:

4. Factoring Perfect Square Trinomials

A perfect square trinomial has the form:

or

Example:

Factor $x^2 + 6x + 9$.

This is a perfect square trinomial because:

5. Factoring the Sum and Difference of Cubes

• Sum of cubes:

  $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$

• Difference of cubes:

  $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$

Example:

Factor $x^3 - 8$.

Since $8 = 2^3$, we can write:

6. Factoring by Grouping

Factoring by grouping is used when you have four terms. Group the terms in pairs, factor each group, and then factor out the common binomial.

Steps:

1. Group terms.
2. Factor out the GCF from each group.
3. Factor out the common binomial.

Example:

Factor $x^3 + 3x^2 + 2x + 6$.

1. Group terms:

   $$(x^3 + 3x^2) + (2x + 6)$$

2. Factor each group:

   $$x^2(x + 3) + 2(x + 3)$$

3. Factor out the common binomial $(x + 3)$:

   $$(x + 3)(x^2 + 2)$$

7. Solving Polynomial Equations by Factoring

After factoring, you can solve polynomial equations by setting each factor equal to zero and solving for the variable.

Example:

Solve $x^2 - 5x + 6 = 0$.

1. Factor the quadratic:

   $$x^2 - 5x + 6 = (x - 2)(x - 3)$$

2. Set each factor to zero:

   $$x - 2 = 0 \quad \text{or} \quad x - 3 = 0$$

3. Solve:

   $$x = 2 \quad \text{or} \quad x = 3$$

Summary of Key Techniques:

• GCF: Always factor this out first.
• Trinomials: Look for two numbers that multiply to $c$ and add to $b$.
• Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$.
• Perfect Square Trinomials: Recognize $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$.
• Cubes: Use formulas for sum and difference of cubes.
• Grouping: Useful for four-term polynomials.

With these techniques, you’ll be able to factor a wide variety of polynomials. Practice is essential, so keep solving different problems to master this skill!