Factoring is a helpful method in algebra that makes working with polynomials easier. By breaking down complicated expressions into simpler parts that you can multiply together, you can solve problems more easily. Let’s go over some important ways to factor polynomials.

1. Greatest Common Factor (GCF)

The first thing you often do when factoring is to find the Greatest Common Factor (GCF). The GCF is the largest number or expression that divides all parts of the polynomial.

Example:

Let’s look at the polynomial $6x^3 + 9x^2$.

Step 1: Find the GCF of the numbers in front ($6$ and $9$), which is $3$.
Step 2: Look for the lowest power of $x$ in both terms, which is $x^2$.
Step 3: Use the GCF to factor the expression:

$6x^3 + 9x^2 = 3x^2(2x + 3)$

2. Difference of Squares

Another useful method is factoring the difference of squares. This works for expressions that look like $a^2 - b^2$, which can be factored into $(a + b)(a - b)$.

Example:

Consider the expression $x^2 - 25$.

Here, we see:

• $a^2 = x^2$ (so, $a = x$)
• $b^2 = 25$ (so, $b = 5$)

Using the difference of squares, we can factor it like this:

$x^2 - 25 = (x + 5)(x - 5)$

3. Factoring Trinomials

Factoring trinomials takes some practice but is very helpful. You look for two numbers that multiply to give you the last number and add to give you the middle number.

Example:

For the trinomial $x^2 + 5x + 6$, we need numbers that multiply to $6$ and add to $5$.

The numbers $2$ and $3$ work! So we can factor the trinomial like this:

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

Conclusion

By using these factoring methods—GCF, difference of squares, and factoring trinomials—you can make working with polynomials easier. Practice these techniques, and soon factoring will feel natural. You’ll be ready to take on algebra problems with confidence!