Lesson 4: Special Products and Factors

New Concept

Certain binomials create predictable patterns when multiplied, called special products. Recognizing these forms—like $(a \pm b)^2$ and $a^2-b^2$—gives you a powerful shortcut for both multiplying and factoring polynomials quickly, without the usual trial and error.

What’s next

Now, we'll break down the formulas for these special products. You'll then master them through worked examples and a series of interactive practice cards.

Property

1. $(a + b)^2 = a^2 + 2ab + b^2$

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

Examples

• To expand $(x + 5)^2$, we use the formula with $a=x$ and $b=5$: $(x)^2 + 2(x)(5) + (5)^2$, which simplifies to $x^2 + 10x + 25$.

Property

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

Examples

• To multiply $(x - 6)(x + 6)$, we identify $a=x$ and $b=6$. The result is $(x)^2 - (6)^2 = x^2 - 36$.

• To multiply $(3p - 5q)(3p + 5q)$, we identify $a=3p$ and $b=5q$. The result is $(3p)^2 - (5q)^2 = 9p^2 - 25q^2$.

Property

1. $a^2 + 2ab + b^2 = (a + b)^2$

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

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

Property

The sum of two squares, $a^2 + b^2$, cannot be factored.

Examples

• The polynomial $x^2 + 100$ cannot be factored because it is the sum of two squares, $x^2$ and $10^2$.

• The polynomial $9y^2 + 16$ cannot be factored because it is the sum of two squares, $(3y)^2$ and $4^2$.