Factoring Solved Examples | Mathematics for Digital SAT PDF Download

Example 1: Algebra

If  , and  , what is the value of ?

(a) 0
(b) -6
(c) 6
(d) 8
(e) –8

 Correct Answer is Option (d)
The numerator on the left can be factored so the expression becomes  which can be simplified to (x - 3) = 5.
Then you can solve for  x by adding 3 to both sides of the equation, so x = 8

Example 2: Factoring

Correct Answer is Option (d)
You need to find two numbers that multiply to give 72 and add up to give 18
easiest way: write the multiples of 72:
1, 72
2, 36
3, 24
4, 18
6, 12: these add up to 18
(x + 6)(x + 12)

Example 3: How To Factor A Variable

Correct Answer is Option (e)
Nothing common cancels at the beginning. To factor this, we need to find two numbers that multiply to 9 x 4 = 36 and sum to 12. 6 and 6 work.
So 9x² + 12x + 4 = 9x² + 6x + 6x + 4
Let's look at the first two terms and last two terms separately to begin with. 9x² + 6x can be simplified to 3x(3x + 2) and 6x + 4 can be simplified into 2(3x + 2). Putting these together gets us
9x² + 12x + 4
= 9x² + 6x + 6x + 4
= 3x(3x + 2) + 2(3x + 2)
= (3x + 2)(3x + 2)
This is as far as we can factor. 

Example 4 : Factoring

Correct Answer is Option (b)
We can first factor out −3:
−3(4x² - 9)
This factors further because there is a difference of squares:
−3(2x + 3)(2x − 3)

Example 5: How To Factor A Variable

Correct Answer is Option (d)
Group all the terms with the x variable.
x³ + 2x² + a + bx² + 2 = (x³ + 2x² + bx²) + a + 2
Pull out an x² term from parentheses.
(x³ + 2x² + bx²) + a + 2 = x²(x + 2 + b) + a + 2
There are no more common factors.
The correct answer is:  x²(x + 2 + b) + a + 2

Example 6: Algebra

Factor and simplify: 
(a) 8y − 12
(b) −4
(c) 8y
(d) 8y − 4
(e) 8y + 4

Correct Answer is Option (d)
64y² − 16  is a difference of squares.
The difference of squares formula is a² − b² = (a − b)(a + b).
Therefore,

= 8y − 4