Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).

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1 Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32 c A factor of a polynomial is a polynomial that will divide the given polynomial evenly (with no remainder). a. The factors of a 3 are 1, a, a 2, and a 3 b. Since 3x(xy 3 ) = 3x 2 y 3, both 3x and xy 3 are factors of 3x 2 y 3. c. Given 2x(3x - 4y) = 6x 2-8xy i. The monomial factor of 6x 2-8xy is ii. The binomial factor of 6x 2-8xy is 3. The greatest common factor (GCF) of two or more integers is the largest integer that is a factor of all of the given integers. Find the GCF of a. 24 an 60 b. 16, 8, and 12

2 Math 50, Chapter 8 (Page 2 of 20) 4. The greatest common factor (GCF) of two or more monomials is the product of the GCF of the coefficients and all common variable factors. Find the GCF of a. 12a 4 b and 18a 2 b 2 c b. 14a 3 and 49a 7 c. 4x 6 y and 18x 2 y 5 z d. 2xy 2, 4xy and 8x e. 2xy 2 (y - 4) and 10xy(y - 4) 2 5. The Distributive Property states a(b + c) = ab + ac, where the product a(b + c) is called the factored form of the expression and ab + ac is called the expanded form of the expression. To expand an expression means to apply the distributive property and write the expression as terms (addends). To factor an expression means to apply the distributive property and write the expression as a product. The Factored form of the expression is written as a product. Expanding / Multiplying a(b+c) = ab+ ac The Expanded form of the expression is written as a sum. Factoring The monomial factor is The binomial factor is

3 Math 50, Chapter 8 (Page 3 of 20) Steps to factor a monomial from a multi-term polynomial 1. Find the GCF of all the terms. 2. Rewrite each term of the polynomial as a product of monomials where one of the factors is the GCF. 3. Factor out the GCF. Write the polynomial in factored form where one of the factors is the GCF. a. Factor 5x 3-35x 2 +10x b. Factor 16x 2 y + 8x 4 y 2-12x 4 y 5 c. Factor 14a 2 x - 21a 4 b d. Factor 6x 4 y 2-9x 3 y x 2 y 4

4 Math 50, Chapter 8 (Page 4 of 20) Note The common factor may be a binomial. e. Factor x(y - 3)+ 4(y - 3) f. Factor a(b- 7) -b(b - 7) Fact a - b = -1(b - a) = -(b - a) Factoring out -1 Factor out -1 from each binomial. a. a - 2b = b. 3x - 2y = c. 4a - 3b = Factor each expression into the product of two binomials. a. a(a - b) + 5(b - a) b. 2x(x - 5) + y(5 - x) b. 3y(5x - 2) - 4(2-5x) c. 5a(x - 4) - (4 - x)

5 Math 50, Chapter 8 (Page 5 of 20) Factor by Grouping - Factoring Expressions with Four Terms 1. Group the first 2 terms and the last two terms. 2. Factor out the GCF from each group. 3. Write the expression as the product of two binomials. Factor. a. 2x 3-3x 2 + 4x - 6 b. 3x 3-4x 2-6x + 8 c. y 5-5y 3 + 4y 2-20 d. 10xy 2-15xy + 6y - 9

6 Math 50, Chapter 8 (Page 6 of 20) 8.2 Factor Trinomials of the form x 2 + bx + c x 2 + bx + c b c Factored form x 2 + 9x +14 (x + 2)(x + 7) x 2 - x -12 (x - 4)(x + 3) x 2-2x -15 (x - 5)(x + 3) Note The leading coefficient (the coefficient on the x 2 term) is one. Factor a Trinomial in the Form x 2 + bx + c To factor a trinomial of the form x 2 + bx + c means to express the trinomial as the product of two binomials. a. (x + 6)(x + 2) = x2 + 8x +12 b. (x - 4)(x - 5) = x2-9x + 20 c. (x - 6)(x + 2) = x2-4x -12 d. (x + 7)(x - 4) = x2 + 3x - 28 Observations 1. The sum of the constant binomial terms equals the linear coefficient in the trinomial. 2. The product of the binomial constant terms is the constant term of the trinomial.

7 Math 50, Chapter 8 (Page 7 of 20) To Factor Trinomials in the form x 2 + bx + c 1. Find two numbers that have a product of c and a sum of b. 2. Write the trinomial as a product of two binomials. That is, use the two numbers from step 1 to fill in the following blanks, x 2 + bx + c = (x + )(x + ) Factor each trinomial a. x 2-2x - 24 b. x x + 32 c. x 2-8x + 15 d. x 2-6x -16 e. x 2 + 3x -18 f. x 2-6x - 8

8 Math 50, Chapter 8 (Page 8 of 20) g. Factor x 2 +17x - 60 h. Factor x 2-21x Factor Completely 1. Factor out any monomial factor that is common to all terms. 2. Factor the trinomial into the product of two binomials. a. Factor 3x x x b. Factor 3a 2 b -18ab - 81b c. Factor x 2 + 9xy + 20y 2 d. Factor x 2-19xy + 84y 2

9 Math 50, Chapter 8 (Page 9 of 20) Example Determine the integer values for b so that the trinomial can be factored. a. x 2 + bx + 30 b. x 2 -bx +18

10 Math 50, Chapter 8 (Page 10 of 20) 8.3 Factoring Polynomials of the form ax 2 + bx + c Recall In section 8.2 the topic was to factor polynomials in the form x 2 + bx + c. That is, the method in section 8.2 only works when the leading coefficient is 1. ax 2 + bx + c a b c a c 10x 2 - x - 3 x 2-4 x 4x 2-27x +18 3x x + 12 The ac-method (Factor by Grouping) 1. Find two numbers that have a product of a c and a sum of b. 2. Re-write the middle term (the bx term) of the trinomial using the numbers from step Factor by grouping. Factor each of the following a. 10x 2 - x - 3 b. 4x 2-27x +18 c. 3x x + 12

11 Math 50, Chapter 8 (Page 11 of 20) The ac-method (Factor by Grouping) 1. Find two numbers that have a product of a c and a sum of b. 2. Rewrite the middle term (the bx term) of the trinomial using the numbers from step Factor by grouping. Factor each of the following d. 6x 2-5x - 6 e. 8x x -15 f. 15-2x - x 2 g. y 2 + 2y - 24

12 Math 50, Chapter 8 (Page 12 of 20) When factoring, always factor completely. 1. Factor out any monomial factor common to all terms. 2. Find two numbers that have a product of a c and a sum of b. 3. Rewrite the middle term (the bx term) of the trinomial using the numbers from step Factor by grouping. Factor a. 3x 3-23x x b. 4a 2 b 2-30a 2 b + 14a 2 c. 9a 3 b - 9a 2 b 2-10ab 3 d. 45a 3 b - 78a 2 b ab 3

13 Math 50, Chapter 8 (Page 13 of 20) 8.4 Factoring The Difference of Two Squares, a 2 - b 2 Factoring the Difference of Two Squares The binomial a + b is the sum of two terms and a - b is the difference of two terms. The difference of two squares, a 2 - b 2 factors into the product of the sum and difference of two terms. That is, a 2 - b 2 = (a + b)(a - b) e.g. x 2-9 = (x) 2 - (3) 2 = (x + 3)(x - 3) 1-64y 2 = (1) 2 - (8y) 2 = (1+ 8y)(1-8y) Factor a. x 2-16 = (x) 2 - (4) 2 = (x + 4)(x - 4) b. a 2-25 = ( ) 2 - ( ) 2 = c. z 4-49 d. c 6-36 e. x 4-10 f. 25c 2 - d 2 g. 6x 2-1

14 Math 50, Chapter 8 (Page 14 of 20) a. Factor x b. Factor 4z 6 +16a 2 When asked to factor, always factor completely. a. Factor p 8-16 b. Factor n 4-81 c. Factor m 4-256

15 Math 50, Chapter 8 (Page 15 of 20) Factoring Perfect-Square Trinomials Definition The square of a binomial is a perfect-square trinomial. Thus a perfect-square trinomial factors into a binomial squared. ( binomial) 2 = Perfect -Square Trinomial 1. (a + b) 2 = a 2 + 2ab + b 2 2. (a - b) 2 = a 2-2ab + b 2 Fact A trinomial is a perfect-square trinomial if 1. The first and last terms are perfect squares, and 2. The middle term is twice the product of the un-squared first and last terms. Determine if each trinomial is a perfect-square trinomial. Then factor each trinomial. a. Factor 9x x +16 = (3x) x + (4) 2 = b. Factor 9x 2-30x + 25 c. Factor 16y 2 + 8y +1

16 Math 50, Chapter 8 (Page 16 of 20) d. Factor 4x x + 9 e. Factor x x + 6

17 Math 50, Chapter 8 (Page 17 of 20) Guidelines to Factor a Polynomial 1. If there is a common monomial factor in all the terms, then factor it out. 2. Is the polynomial one of the special forms? a. The difference of two squares: a 2 - b 2 = (a + b)(a - b). b. Perfect-square trinomials factor into a binomial-squared. i. a 2 + 2ab + b 2 = (a + b) 2 ii. a 2-2ab + b 2 = (a - b) 2 3. If the polynomial contains four terms, then try factor by grouping. 4. If the polynomial is a trinomial with a leading coefficient of 1 (i.e. x 2 + bx + c) then it factors into the product of two binomials: x 2 + bx + c = (x + )(x + ). The constant terms in the binomials have a product of c and a sum of b. 5. If the polynomial is of the form ax 2 + bx + c, then factor using the ac-method. Find the two numbers that have a product of ac and a sum of b. Use those numbers to rewrite the bx term as two terms. Then factor the four term polynomial by grouping. Factor a. 3x 2-48 b. x 3-3x 2-4x+12 c. 12x 3-75x d. a 2 b - 7a 2 - b + 7

18 Math 50, Chapter 8 (Page 18 of 20) 8.5 Solving Equations Definitions A second degree polynomial equation of the form ax 2 + bx + c = 0 is called a quadratic equation in standard form. Furthermore, the ax 2 term is called the quadratic term. The linear term is bx. The constant term is c. Principle of Zero Products If the product of two factors is zero, then at least one of the factors must be zero. If a b = 0, then a = 0 or b = 0.. Solve a. (x - 2)(x - 11) = 0 b. (x - 12)(x + 6) = 0 c. (2x + 3)(4 x - 1)(5x + 11) = 0 d. (3x - 4)(2x + 1)(6x - 7) = 0 Note To solve equations using the principle of zero products both of the following conditions must be met. 1. One side of the equation must be zero. 2. The other side of the equation must be in factored form.

19 Math 50, Chapter 8 (Page 19 of 20) Solve a. 2x 2 + x = 6 b. 2y 2 - y = 1 c. 2x 2 = 50 d. 16a 2-49 = 0 e. x 2-8x = 0 f. 3a 2 = a g. (x - 3)(x - 10) = -10 h. (x + 2)(x - 7) = 52

20 Math 50, Chapter 8 (Page 20 of 20) Application Problems Example 1 The sum of the squares of two consecutive positive odd integers is equal to 130. Find the two integers. Example 2 A stone is thrown into a well with an initial velocity of 8 ft/s. The well is 440 ft deep. How many seconds later will the stone hit the bottom of the well? Use the equation d = vt +16t 2, where d is the distance in feet, v is the initial velocity in feet per second, and t is the time in seconds. Problem 3 The length of a rectangle is 3 m more than twice the width. The area of the rectangle is 90 m 2. Find the length and width of the rectangle.