Factoring Polynomials

Factoring a polynomial means writing it as a product of simpler polynomials. It is the reverse process of multiplying polynomials together. Here are the main techniques you need to know to break down polynomials completely.

1. Greatest Common Factor (GCF)

Always look for the GCF first. The GCF is the largest numerical and variable expression that divides evenly into every term of the polynomial.

Example: Factor $8x^3 + 12x^2$. The largest number that divides 8 and 12 is 4. The highest power of $x$ shared by both terms is $x^2$. $8x^3 + 12x^2 = 4x^2(2x + 3)$

2. Factoring by Grouping

This method is often used for polynomials with four terms. You group the terms into pairs, factor out the GCF from each pair, and then factor out the common binomial.

Example: Factor $x^3 + 2x^2 + 3x + 6$. Group the terms: $(x^3 + 2x^2) + (3x + 6)$. Factor each group: $x^2(x + 2) + 3(x + 2)$. Factor out the common binomial $(x + 2)$: $(x^2 + 3)(x + 2)$

3. Factoring Trinomials

Form $x^2 + bx + c$

When the leading coefficient is 1, find two numbers that multiply to $c$ and add to $b$.

Example: $x^2 + 5x + 6$. The numbers that multiply to 6 and add to 5 are 2 and 3. Result: $(x + 2)(x + 3)$.

Form $ax^2 + bx + c$

When the leading coefficient $a$ is not 1, use the "ac method" to split the middle term and factor by grouping.

Example: Factor $6x^2 - 7x - 20$.

1. Multiply $a$ and $c$: $6 \times (-20) = -120$.
2. Find factors of $-120$ that add to $-7$. Those numbers are $-15$ and $8$.
3. Rewrite the middle term: $6x^2 - 15x + 8x - 20$.
4. Factor by grouping: $3x(2x - 5) + 4(2x - 5)$. $(3x + 4)(2x - 5)$

4. Special Patterns

Recognizing common patterns can save you a lot of time. The most important one is the Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$

Example: Factor $x^4 - 16$. Rewrite the expression as a difference of squares: $(x^2)^2 - 4^2$. Factor using the pattern: $(x^2 - 4)(x^2 + 4)$. Notice that $x^2 - 4$ is another difference of squares! Factor it completely: $(x - 2)(x + 2)(x^2 + 4)$ (Note: $x^2 + 4$ is a sum of squares and cannot be factored further using real numbers).