Factoring Techniques

When you're looking at polynomial expressions, grouping is a powerful strategy that works with 4 terms. The key is finding patterns that let you rewrite the expression more simply.

For example, with expressions like $x^3+2x^2-3x-6$, you can factor by grouping: $x(x+2)-3(x+2) = (x+2)(x^2-3)$. Notice how $(x+2)$ appears in both parts, making it a common factor.

Difference of squares is another important pattern: $4x^2-81 = (2x+9)(2x-9)$. Remember: $a^2-b^2 = (a+b)(a-b)$. For perfect cubes, remember these formulas:

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

Pro Tip: When facing complex expressions, try factoring out the greatest common factor first, then look for patterns like difference of squares or perfect cubes.

Let's see factoring in action with a word problem. A landscape architect needs to design a marble planter holding 4 cubic feet of soil. If length = 6x height, width = 3x height, and sides are 1 foot thick, we can use factoring to find the dimensions: $18x^3-18x^2+4x-4=0$. After factoring to $(x-1)(18x^2+4)=0$, we get $x=1$, making the outer dimensions 1ft height, 3ft width, and 6ft length.