ALGEBRA REVIEW - FACTORING and OTHER HINTS

Suggestion – try the examples with a pencil

Step 1. Look for Common factors – take outside the parentheses

ex. 12x⁴yz² − 32x³y⁴z⁴
= 4x³yz²(3x − 8y³z²)

Step 2. Look for Special Cases

Binomials – Difference of Squares a² − b² = (a + b)(a − b)

ex. 9p² − 25q²
= (3p + 5q)(3p − 5q)

Trinomials − Perfect Square a² + 2ab + b² = (a + b) ²

ex. 4x² − 28xy + 49y²
= (2x − 7y)²

Step 3. No special case Trinomials ax² + bx + c Two methods

ex. 6x² − x − 12

Systematic Method

1. Multiply coefficients of first and last terms:

a * c
(6)(−12) = − 72

2. Look for the factors of ac ( coefficient of first
and last terms) that add to middle
term coefficient.

and −9 + 8 = −1

3. Re-write ex pression breaking middle term
using those factors.

6x² − 9x + 8x − 12

4. Factor again by grouping

= 3x(2x − 3) + 4(2x − 3)
= (2x − 3)(3x + 4) This comes out the same answer as the Guess and Check method.

Guess and Check Method

This method is perfectly
acceptable and often faster.
Simply write down any factors
of first and last terms- each in a
set of parentheses.
1. Revise Tries
2. Watch Signs.
3. Check by using FOIL

Try 1. (3x – 4)(2x + 3) doesn’t work
Switch signs

Try 2. (3x + 4)(2x − 3) works

Cubics

a³ − b³ = (a − b)(a² + ab + b²) ex. 8x³ − 27y³ = (2x − 3y)(4x² + 6xy + 9y²)
a³ + b³ = (a + b)(a² − ab + b²)

Polynomials – Factor by Grouping Look for ways to group if possible.

ex. − 4b + 9a² − b² − 4 Notice the terms with the variable ‘b’ --Clump them together.
= 9a² − b² − 4b − 4 Put parentheses around terms with ‘b’ -- watch negative sign.
= 9a² − (b² + 4b + 4)
= 9a² − (b + 2) ² This is the difference of two perfect squares.
= [3a + (b + 2)][3a − (b + 2)]
= (3a + b + 2)(3a − b − 2) ALWAYS CHECK

Negative signs – how to deal with them – using the function notation f(–x)

Given a function: f(x) = (x − 2)² Find f(–x)

If you replace x with –x , you are increasing your chances of making an error. For example: you would
get (–x –2)² For me, this is a lot of negatives. It’s (–x –2) (–x –2). Why not multiply out the original
problem first. i.e. (x –2)² = x²–4x + 4 THEN replace the x with –x.

You get (–x) ²– 4(–x) + 4 = x² + 4x + 4

Also, if you have something like – x² + x +6, sometimes it helps to factor out the negative and then refactor:

e.g. –(x² – x – 6) = – (x –3)(x + 2)

This is a lot easier to deal with than trying to factor – x² + x– 6 . If you want just two factors, then you can
bring through the negative:

e.g. – (x –3)(x + 2) = (3– x)(x +2)

Be careful. You only need to bring the negative through one of the factors.
WHY? – A*B = A*–B NOT –A*–B

OTHER HINTS

• Always write the original problem and don’t touch it.
• Write it again if you want to revise or transform it.

Solving a problem

• Combine like terms before you do anything else so the variable is in only one term.
• Always ISOLATE the term with the variable for which you are solving. That is, get
everything away from the term with the variable.
• If you have fractions, don’t do anything until the term with the variable is isolated. Then
multiply both sides of the equation by LCD.

Combinations of Different Types of Equations – Worksheet OPTIONAL

1. 20x³ =125x Put all x’s on one side and factor

2. x³ + 2x² + 4x + 8 =0 Factor and solve

3. Quadratic Look –alikes Let t² = x and rewrite as a regular quadratic -CHECK
36t ⁴+24t ² − 7 = 0

4. Radical Isolate radical and square both sides CHECK

5. This is a cube root so Cube both sides CHECK

6. Rational exponents - raise to a power that make the side with the variable a power of 1

7. Complex Fractions – Simplifying – not solving. To Simplify-- Multiply numerator and denominator
by LCD of both. Combine terms – watch negative.

Here, the LCD is (t + 2)(t − 3)

8. Fractional - You do not need to put OVER LCD since this is an equation. Just
multiply both sides by the LCD of both sides.

9. Absolute Value – Use property of absolute value function