Section 5 4 Factoring FACTORING Greatest Common Factor

Section 5. 4 Factoring FACTORING Greatest Common Factor, Factor by Grouping, Factoring Trinomials, Difference of Squares, Perfect Square Trinomial, Sum & Difference of Cubes 1

Factoring—define factored form Factor means to write a quantity as a multiplication problem v a product of the factors. l Factored forms of 18 are: l 2

Factoring: The Greatest Common Factor l 1. 2. 3. l 3 To find the greatest common factor of a list of numbers: Write each number in prime factored form Choose the least amount of each prime that occurs in each number Multiply them together Find the GCF of 24 & 36

Factoring: The Greatest Common Factor l 1. 2. 3. l 4 To find the greatest common factor of a list of variable terms: Choose the variables common to each term. Choose the smallest exponent of each common variable. Multiply the variables. Find the GCF of:

Factoring: The Greatest Common Factor l 1. 2. 3. 5 To factor out the greatest common factor of a polynomial: Choose the greatest common factor for the coefficients. Choose the greatest common factor for the variable parts. Multiply the factors.

Factoring: The Greatest Common Factor each of the following by factoring out the greatest common factor: l 6 5 x + 5 = 4 ab + 10 a 2 = 8 p 4 q 3 + 6 p 3 q 2 = 2 y + 4 y 2 + 16 y 3 = 3 x(y + 2) -1(y + 2) =

Factoring: The Greatest Common Factor l 7 The answers are :

Factoring: by Grouping l 1. 2. 3. ¥ 8 Often used when factoring four terms. Organize the terms in two groups of two terms. Factor out the greatest common factor from each group of two terms. Factor out the common binomial factor from the two groups. Rearranging the terms may be necessary.

Factoring: by Grouping l Factor by grouping: 1. 2 groups of 2 terms Factor out the GCF from each group of 2 terms 2. Factor out the common binomial factor 3. 9

Factoring: by Grouping Factor by grouping 10

Factoring Trinomials—with a coefficient of 1 for the squared term l 1. 2. 3. 4. 11 Factor: List the factors of 20: Select the pairs from which 12 may be obtained Write the two binomial factors: Check using FOIL:

Factoring Trinomials 12 TIP If the last term of the trinomial is positive and the middle sign is positive, both binomials will have the same “middle” sign as the second term.

Factoring Trinomials 13 TIP If the last term of the trinomial is positive and the middle sign is negative, both binomials will have the same “middle” sign as the second term.

Factoring Trinomials—with a coefficient of 1 for the squared term l 1. 2. 3. 4. 14 Factor List the factors of 22 Select the pair from which – 9 may be obtained Write the two binomial factors: Check using FOIL:

Factoring Trinomials TIP 15 If the last term of the trinomial is negative, both binomials will have one plus and one minus “middle” sign.

Factoring Trinomials—primes A PRIME POLYNOMIAL cannot be factored using only integer factors. l Factor : l The factors of 5: 1 and 5. l Since – 2 cannot be obtained from 1 and 5, the polynomial is prime. l 16

Factoring Trinomials— 2 variables Factor: l The factors of 8 are: 1, 8 & 2, 4, & -1, -8 & -2, -4 l Choose the pairs from which – 6 can be obtained: 2 & 4 l Use y in the first position and z in the second position l Write the two binomial factors and check your answer l 17

Factoring Trinomials—with a GCF l l 18 If there is a greatest common factor? If yes, factor it out first.

Factoring Trinomials—always check your factored form l 19 Always check your answer with multiplication of the factors. l The check:

Factoring Trinomials— when the coefficient is not 1 on the squared term l 20

Factoring Trinomials---use grouping l 21

Factoring Trinomials---use grouping l 22

Factoring Trinomials---use FOIL and Trial and Error l 23

Factoring Trinomials---use FOIL and Trial and Error l 24

Factoring Trinomials---use FOIL and Trial and Error l 25

Factoring Trinomials---use FOIL and Trial and Error l 26

Factoring Trinomials---use FOIL and Trial and Error l 27

Factoring Trinomials---use FOIL and Trial and Error l 28

Factoring Trinomials---with a negative GCF l l l 29 Is the squared term negative? If yes, factor our a negative GCF.

Special Factoring—difference of 2 squares ü The following must be true: 1. There must be only two terms in the polynomial. Both terms must be perfect squares. There must be a “minus” sign between the two terms. 2. 3. 30

Special Factoring—difference of 2 squares l The following pattern holds true for the difference of 2 squares: l 31

Special Factoring—difference of 2 squares l l 32 The pattern:

Special Factoring—difference of 2 squares l l 33 The pattern:

Special Factoring—difference of 2 squares l l 34 The pattern:

Special Factoring—difference of 2 squares l l 35 The pattern:

Special Factoring—perfect square trinomial l l 36 A perfect square trinomial is a trinomial that is the square of a binomial.

Special Factoring—perfect square trinomial l l Ø 37 The first and third terms are perfect squares. AND the middle term is twice the product of the square roots of the first and third terms TEST THE MIDDLE TERM:

Special Factoring—perfect square trinomial l 38 The patterns for a perfect square trinomial are:

Special Factoring—perfect square trinomial l 39 Factor the following using the perfect square trinomial pattern:

Special Factoring—perfect square trinomial l 40 Factor the following using the perfect square trinomial pattern:

Special Factoring—difference of two cubes l 41 Factor using the pattern.

Special Factoring—sum of two cubes l 42 Factor using the pattern.

Solving quadratic equation with factoring l. A quadratic equation has a “squared” term. 43

ZERO FACTOR PROPERTY To Factor a Quadratic, Apply the Zero-Factor Property. Ø If a and b are real numbers and if ab = 0, then a = 0 or b = 0. 44

Solving quadratic equations with factoring—Zero-Factor Property l Solve the equation: (x + 2)(x - 8) = 0. l Apply the zero-factor property: (x + 2) = 0 x = -2 45 or or (x – 8) = 0 x=8

Solving quadratic equations with factoring—Zero-Factor Property There are two answers for x: -2 and 8. l Check by substituting the values calculated for x into the original equation. (x + 2)(x - 8) = 0. (-2 + 2)(-2 – 8) = 0 (8 + 2)(8 – 8) = 0 0=0 l 46

Solving quadratic equations with factoring—Standard Form l 1. 47 To solve a quadratic equation, Write the equation in standard form. Ø (Solve the equation for 0. )

Solving quadratic equations with factoring l 2. 48 To solve a quadratic equation, Factor the quadratic expression.

Solving quadratic equations with factoring l 3. 49 To solve a quadratic equation, Apply the Zero-Factor Property

Solving quadratic equations with factoring l 4. 50 To solve a quadratic equation, Check your answers