Factoring

To factor an algebraic expression we write the algebraic expression as a product of its factors

(1) Finding the greatest common factor: the largest number that divides evenly into each term

Examples:

(a) Factor: 5x is the greatest common factor

Divide by 5x

(b) Factor: is the greatest common factor

(c) Factor: y – 2 is the greatest common factor

In factoring any algebraic expression, the first step is always to look for a

Greatest common factor.

(2) The Difference of Two Squares

Noting that . A binomial of the form has for its

factors

Examples:

(a) Factor:

(b) Factor:

(c) Factor:

(d) Factor:

is prime. Only factors are 1 and

(3) Trinomials of the Form

Noting that : we factor trinomials of the form

by:

(1) Finding 2 numbers whose product is c and whose sum is b

(2) Writing them in the form

Examples:

(a) Factor:

Two numbers whose product is 6 and sum is 5: 3 and 2

(b) Factor:

Two numbers whose product is 12 and sum is -8: -6 and -2

(c) Factor:

Two numbers whose product is -54 and sum is -3: -9 and 6

(4) Trinomials of the form

Noting that : We factor trinomials of the form

by:

(1) Writing all pairs of factors of a

(2) Write all pairs of factors of c

(3) Try different combinations of factors of a and c so that the sum of the products of the inner terms and outer terms is bx

Examples:

(a) Factor:

Step 1: Factors of 4: (1)(4), (2)(2)

Step 2: Factors of 3: (1)(3), (-1)(-3)

Since both 3 and 7 are positive we only use the positive factors

Step 3:

(b) Factor:

Step 1: Factors of 5: (1)(5)

Step 2: Factors of 3: (1)(3), (-1)(-3)

Since 3 is positive and 12 is negative we only use the negative

factors of 3

Step 3:

(c) Factor:

Step 1: Factors of 6: (1)(6), (2)(3)

Step 2: Factors of 12: (1)(-12), (-1)(12)

(2)(-6), (-2)(6)

(3)(-4), (-3)(4)

Step 3:

Problems

Factor

(1) (7)

(2) (8)

(3) (9)

(4) (10)

(5) (11)

(6) (12)

Answers

(1) (7)

(2) (8)

(3) (9)

(4) (10)

(5) (11)

(6) (12)

Quadratic Equations

A quadratic equation is an equation of the form:

To Solve a Quadratic Equation by Factoring

(1) Set one side of the equation equal to zero

(2) Factor the non-zero side of the equation

(3) Set each factor equal to zero and solve

Examples:

(a) Solve:

(b) Solve:

If a quadratic equation () can not be solved by factoring, we can use the quadratic formula:

Examples:

(a) Solve:

a = 1, b = 3, c = 1

(b) Solve:

a = 2, b = -4, c = -3

Simplify the radical

Problems

Solve

(1) (4)

(2) (5)

(3) (6)

Answers

(1) x = -3, x = -2 (4)

(2) x = -5, x = 7 (5)

(3) (6)