# Factoring Perfect Square Trinomials?

In my Algebra1 class we are SUPPOSED to be learning. Unfortunately, my class is completely obnoxious and we don't seem to accomplish anything but pissing off the teacher. We didn't get through our notes and I have no idea how to factor perfect square trinomials! Can anyone help me? My first problem in that section is

x^2-20x+100

.

The formula to get the answer is

a^2-2ab+b^2

but the answer has to look like (a+b)^2 and I am definetly not getting that as an answer.

Relevance

x² - 20x + 100

Notice the following:

x² = x * x = (x)²

-20x = -2 * 10 * x

100 = 10 * 10 = (10)²

Notice that the middle term is -2 times the square root of the first term times the square root of the last term.

-20x = -2 * √(x²) * √100

When the middle term is +/- 2 times the square root of the first term times the square root of the last term, you have a perfect square binomial.

Given: x² - 20x + 100 = (x)² - 2(x)(10) + (10)²

Means: a = x, b = 10

Plug these into the formula.

a² - 2ab + b² = (a - b)²

x² - 20x + 100 = (x)² - 2(x)(10) + (10)² = (x - 10)²

~~~~~~~~~~~~~~~~

NOTE: If the above problem had had + 20x for the middle term, the answer would have been (x + 10)² because:

Given: x² + 20x + 100 = (x)² + 2(x)(10) + (10)²

Means: a = x, b = 10

Since the middle term is positive, the sign is also positive in the factored form.

x² + 20x + 100 = (x)² + 2(x)(10) + (10)² = (x + 10)²

~~~~~~~~~~~~~~~~

Sometimes you'll come across polynomials that match a particular pattern.

HINT: Memorize these commonly occurring factoring formulas

Perfect square binomials:

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

Difference of two perfect squares:

(a - b)(a + b) = a² - b²

Sum of two perfect cubes:

a³ + b³ = (a + b)(a² - ab + b²)

Difference of two perfect cubes:

a³ - b³ = (a - b)(a² + ab + b²)

Notice that the formulas for the perfect cubes are almost the same, but the signs are different. You can use the SOAP mnemonic to remember the signs.

S = The first sign is always the SAME as the sign between the 2 cubes

O = The second sign is always the OPPOSITE as the sign between the 2 cubes

AP = The last sign is ALWAYS POSITIVE

• Just identify "a" and "b" and plug 'em in. 4x²+20x+25 --> a = 2x and b=5 because (2x)² = 4x² and 5² = 25 and so (a + b)² = (2x + 5)² x²–8x+16 --> Here a = x and b = 4 and we have a MINUS in the middle, and so (x – 4)² and finally here a = 3t and b = 1, so 9t² + 6t + 1= (3t + 1)² This is a rigged exercise. In "the wild" you need to also check that 2ab gives the middle term. It just says multiply the two terms and double it. So for (2x + 5)² we would check that 2(2x)(5) = 20x which it does!