The number part of a term with a variable part. Terms that have the same variable parts. Constant terms are also like terms.

Transcription

1 Algebra Notes Section 9.1: Add and Subtract Polynomials Objective(s): To be able to add and subtract polynomials. Recall: Coefficient (p. 97): Term of a polynomial (p. 97): Like Terms (p. 97): The number part of a term with a variable part. The parts of an expression that are added together. Terms that have the same variable parts. Constant terms are also like terms. Vocabulary : number variable I. Monomial A,, or the product of a number and one or more variables with whole number exponents. sum exponents variables II. Degree of a Monomial: The of the of the in the monomial. monomial sum III. Polynomial: A or a of monomials, each called a term of the polynomial. greatest degree of its terms IV. Degree of a Polynomial: The. V. Leading Coefficient: When a polynomial is written so that the exponents of a variable decrease from left to right, the leading coefficient is the. Rewriting a polynomial so that the exponents of a variable decrease from left to right is often referred to as writing a polynomial in descending order of exponents. Example: 2x 3 + x 2 5x + 12 This polynomial has terms. VI. Binomial: A polynomial with terms. VII. Trinomial: A polynomial with terms. The leading coefficient is. The degree is. The constant term is. VIII. Adding Polynomials: To add polynomials,. 2 3 add like terms coefficient of the first term IX. Subtracting Polynomials: To subtract polynomials, add its opposite (multiply each term by 1) 2

2 Examples: Notes Consider the polynomial 3x 3 4x 4 + x 2. a. What is the degree of the polynomial? 4 b. How many terms does this polynomial have? 3 c. Classify the polynomial according to the number of terms. trinomial d. Rewrite the polynomial in descending order of exponents. 4x 4 + 3x 3 + x 2 e. What is the leading coefficient of the polynomial? 4 f. List all of the coefficients of this polynomial. 4, 3, 1 g. List the terms of the polynomial. 4x 4, 3x 3, x 2 2. Tell whether the expression is a polynomial. If it is, find its degree and classify it by the number of its terms. Otherwise, tell why it is not a polynomial. Expression 4x 2x + 3x m 2 + m x xy + 3x 2 y No. Is it a polynomial? Yes Yes Exponent must be a whole number No. Exponent cannot be negative. Yes Classify by degree and number of terms 1 st degree monomial 5 th degree trinomial 3 rd degree binomial

3 Notes 9.1 page 3 3. Find the sum or difference. a. ( 2x 2 + 3x x 3 ) + (3x 2 + x 3 12) x 2 + 3x 12 b. (4x 3 + 2x 2 4) + (x 3 3x 2 + x) 5x 3 x 2 + x 4 c. (2m 2 8) (3m 2 4m + 1) m 2 + 4m 9 d. (5y 2 + 2y 4) ( y 2 + 4y 3) 6y 2 2y 1 4. During the period , the number of hours an individual person watched broadcast television B and cable and satellite television C can be modeled by B = 2.8t 2 35t and C = 5t t + 712, where t is the number of years since a. Write a polynomial that represents the total number of hours of broadcast and cable watched. B + C = 2.2t t b. About how many hours did people watch in 2002? 2002 is 3 years since 1999, so t = 3 If t = 3, then B + C = 2.2(3) (3) = hours

4 Algebra Notes Section 9.2: Multiply Polynomials Objective(s): To be able to multiply polynomials. Vocabulary : I. Recall properties of multiplying and adding expressions: Examples: 2x 4x = 2x + 4x = 2x 3x 2 = 2x + 3x 2 = 2x 2 y 3 + 4y 3 x 2 = 2x 3 y 2 + 4y 3 x 2 = 2x(4x + 1) = 8x 2 6x 6x 3 3x 2 + 2x 6x 2 y 3 2x 3 y 2 + 4x 2 y 3 8x 2 + 2x II. FOIL Pattern: O F (2x + 3)(4x + 1) = I L F O I L 8x 2 + 2x + 12x + 3 = 8x x + 3 Examples: 1. Find the product. a. 3x 2 (2x 3 x 2 + 4x 3) b. (x + 4)(2x 1) 6x 5 3x x 3 9x 2 2x 2 + 7x 4 c. (2x 1)(3x 4) d. (4x + 3)(x + 2) 6x 2 11x + 4 4x x + 6 e. (x 2 x 2)(3x 1) 3x 3 4x 2 5x + 2

5 Notes Perform the indicated operation. a. (2x + 1) + (3x 2) b. (2x + 1)(3x 2) 5x 1 6x 2 x 2 3. A rectangle has dimensions x + 3 and x + 5. Which expression shows the area of the rectangle? A. x B. x 2 + 3x + 15 C. x 2 + 8x + 1 D. x 2 + 8x E. None of these A = length x width =(x + 3)(x + 5) = x 2 + 8x A rectangular trivet has a ceramic center and a wooden border. The dimensions of the center and border are shown in the diagram. a. Write a polynomial that represents the total area of the trivet. x inches 8 A = length x width = (2x + 8) (2x + 6) = 4x x + 48 x inches 6 b. What is the total area of the trivet if the width of the border is 2 inches? A = 4(2) (2) + 48 = 120 in 2 5. Write a polynomial that represents the area of the shaded region. A = (2x - 1) (x + 2) 10 8 = 2x 2 + 4x x 2 80 = 2x 2 + 3x x + 2 2x - 1

6 Algebra Notes Section 9.3: Finding Special Products of Polynomials Objective(s): To use special product patterns to multiply polynomials. Vocabulary : I. Square of a Binomial Pattern: (a + b) 2 = (write this on your formula sheet) a 2 + 2ab + b 2 (a b) 2 = a 2 2ab + b 2 II. Sum and Difference Pattern: (a + b)(a b) = (write this on your formula sheet) a 2 b 2 Examples: 1. Find the product. a. (2x + 5) 2 b. (3x y) 2 c. (x + 3)(x 3) 4x x x 2 6xy + y 2 x 2 9 d. (4x + y)(4x y) e. (x + 1)(x + 1) f. (2x 1)(2x 1) 16x 2 y 2 x 2 + 2x + 1 4x 2 4x Which special product pattern results in the following polynomial? a. x 2 + 6x + 9 (x + 3) 2 b. x 2 25 (x + 5)(x 5) c. x 2 8x + 16 (x 4) 2

7 Notes Use a special products pattern to find the product without a calculator: = (20 1)(20 + 1) = = Use a special products pattern to find the product without a calculator: = (20 + 1) 2 = = In dogs, the gene E is for straight pointy ears and the gene e is for pointy but droopy ears. Any gene combination with an E results in straight pointy ears on a dog. The Punnett square shows the possible gene combinations of the offspring and the resulting type of ear. E e a. What percent of the possible gene combinations of the offspring result in droopy ears? 25% E EE Straight Ee Straight b. How can a polynomial model the possible combinations of the offspring? e Ee Straight ee Droopy (0.5E + 0.5e) 2 = 0.25E Ee e 2 The coefficient of e 2 shows that 25% of the possible gene combinations result in droopy ears.

8 Algebra Notes Section 9.4: Solve Polynomial Equations in Factored Form Objective(s): To solve polynomial equations. Vocabulary : a b = 0 a = 0 b = 0 I. Zero-Product Property: Let a and b be real numbers. If then or. II. Roots: The solutions to ab = 0. product III. Factoring: Writing a polynomial as a of other polynomials. IV. Greatest Common Monomial Factor (GCF): A with an coefficient that monomial integer divides evenly into each of the polynomial s terms. V. Projectile: An object that is propelled into the air but has no power to keep itself in the air. VI. Vertical Motion Model: (write this on your formula sheet) The height h (in feet) of a projectile can be modeled by the equation h = 16t 2 + vt + s where t is the time (in seconds) the object has been in the air, v is the initial vertical velocity (in feet per second), and s is the initial height (in feet). factor VII. To solve a polynomial You may need to the polynomial, or write it as a product of other equation using the zero-product property: polynomials. Look for the GCF of the polynomial's terms. Examples: 1. Solve each of the following. a. (x + 3)(x 5) = 0 b. (2x + 1)(x + 4) = 0 3 or 5 ½ or 4 2. Name the greatest common monomial factor of the polynomial. a. 8xy + 20x b. 10x 2 y 3 15xy 4x 5xy

9 Notes Factor out the greatest common monomial factor. a. 8x + 12y b. 5x + 10y c. 14x 2 y y 4 x 3 4(2x + 3y) 5(x + 2y) 7x 2 y 2 (2 + 3y 2 x) d. 8x x e. 4x 2 y 5xy + xy 2 f. 27x 2 y x 3 y (4x 3 + 5x 4 + 1) xy(4x 5 + y) 3(9x 2 y 3 + 6x 3 y 2 + 3) 4. Solve. a. 3x x = 0 b. 4x 2 + 2x = 0 3x(x + 6) = 0 2x(2x + 1) = 0 x = 0 or x = 6 x = 0 or x = ½ c. 4x 2 = 14x d. 6x 2 = 15x 4x 2 14x = 0 6x 2 15x = 0 2x(2x 7) = 0 3x(2x 5) = 0 x = 0 or x = ⁷ ₂ x = 0 or x = ⁵ ₂ 5. A dolphin jumped out of the water with an initial velocity of 32 feet per second. After how many seconds did the dolphin enter the water? h = 16t 2 + vt + s h = 16t t 0 = 16t t 0 = 16t(t 2) t = 0 or t = 2 2 seconds

10 Algebra Notes Section 9.5: Factor x 2 + bx + c Objective(s): To factor trinomials of the form x 2 + bx + c. Vocabulary : Note: The method taught to you in this section only applies to a trinomial where the leading coefficient is 1 (ex: x 2 + 5x + 6). You cannot use this method if the leading coefficient is not 1. ( ex: 4x 2 + 8x + 3 ) I. Factoring x 2 + bx + c: (x + p)(x + q) p + q = b p q = c x 2 + bx + c = provided and. Examples: 1. Which of the following trinomials can be factored using the method of this section. Circle all that apply. A. x 2 + 6x + 7 B. 6x 2 + 7x + 1 C. 4x 2 + 7x 2 D. 3x 2 + 4x + 1 E. x 2 3x 4 2. Factor each of the following. a. x x + 18 b. x 2 + 5x + 6 c. x 2 9x + 20 (x + 9)(x + 2) (x + 3)(x + 2) (x 5)(x 4) d. x 2 6x + 8 e. x 2 + 2x 15 f. x 2 5x + 6 (x 2)(x 4) (x + 5)(x 3) (x 2)(x 3) g. x 2 + 3x 10 (x + 5)(x 2)

11 3. Factor each of the following. Notes 9.5 a. x 2 + 5x + 6 b. x 2 x 6 c. x 2 + x 6 d. x 2 5x + 6 (x + 3)(x + 2) (x 3)(x + 2) (x + 3)(x 2) (x 3)(x 2) 4. Study the factoring patterns in # 3. a. What happens with the factors (p and q) when you have the + + pattern (part a)? Both p and q are positive numbers. b. What happens with the factors (p and q) when you have the pattern (part b)? One is positive the other is negative. The bigger number must be negative. c. What happens with the factors (p and q) when you have the + pattern (part c)? One is positive the other is negative. The bigger number must be positive. d. What happens with the factors (p and q) when you have the + pattern (part d)? Both p and q are negative numbers. 5. Solve the equation. a. x 2 + 3x = 18 b. x 2 2x = 24 c. x 2 = 3x + 28 (x +6)(x 3) = 0 (x 6)(x + 4) = 0 (x 7)(x + 4) = 0 x = 6 or x = 3 x = 6 or x = 4 x = 7 or x = 4 6. You are designing a flag for the school football team with the dimensions shown in the diagram. The shaded region will show the team name. The flag requires 117 square inches of fabric. Find the width w of the flag. w w(w + 4) = 117 w 2 + 4w 117 = 0 (w + 13)(w 9) = 0 2 " w + 2 w = 13 or w = 9. w = 13 does not make sense with the problem. Therefore, the width must be 9.

12 Algebra Notes Section 9.6: Factor ax 2 + bx + c Objective(s): To factor trinomials of the form x 2 + bx + c. Vocabulary : I. Two methods for factoring ax 2 + bx + c: 1. Guess and check with factors of a and c: Factor 2x 2 7x + 3 Factors of 2 Factors of 3 Possible Factorization Middle term multiplied 1, 2 1, 3 1, 2 3, 1 (x 1)(2x 3) 3x 2x = 5x (x 3)(2x 1) x 6x = 7x Answer: (x 3)(2x 1) 2. Grouping method: Factor 3x x 5 Step 1: Find two numbers whose product is: and whose sum is: a c b product must be ( 5)(3) = 15 and the sum must be and 1 work. 15( 1) = 15 and = 14 Step 2: Rewrite the middle term, 14x, using the two numbers you found in step 1. You will have a polynomial with four terms. 3x x 5 3x x x 5 Step 3: Group the first two terms and factor; group the last two terms and factor. There should be an common binomial factor in each of these. Factor the common binomial from each term. 3x(x + 5) 1(x + 5) (x + 5)(3x 1) II. Factoring when a is negative: To factor a trinomial of the form ax 2 + bx + c when a is negative, first factor 1 from each term of the trinomial. Then factor the resulting trinomial using either guess and check or grouping.

13 Examples: Notes Factor any two of the following by guess and check and factor the other two by grouping. a. 2x 2 13x + 6 b. 4x 2 12x 7 c. 3x 2 + 8x + 4 d. 4x 2 9x + 5 (x 6)(2x 1) (2x + 1)(2x 7) (x + 2)(3x + 2) (x 1)(4x 5) 2. Factor each of the following using any method. a. 4x x + 7 b. 3x 2 x + 2 c. 3x 2 13x + 4 (2x +1)(2x 7) (x + 1)(3x 2) (3x + 1)(x + 4) 3. A soccer goalie throws a ball into the air at an initial height of 8 feet and an initial vertical velocity of 28 feet per second. a. Write an equation that gives the height (in feet) of the soccer ball as a function of the time (in seconds) since it left the goalie s hand. h = 16t 2 + vt + s h = 16t t + 8 b. After how many seconds does it hit the ground? 0 = 4(4t 2 7t 2) 4(4t + 1)(t 2) = 0 t = ¼ or t = 2 The ball hits the ground after 2 seconds. 4. A rectangle s length is 5 feet more than 4 times the width. The area is 6 square feet. What is the width? w(4w + 5) = 6 4w 2 + 5w 6 = 0 (4w 3)(w + 2) = 0 w = ¾ or w = 2 w = ¾ ft

14 Algebra Notes Section 9.7: Factor Special Products Objective(s): To factor special products. Vocabulary : I. Difference of Squares Factoring Pattern: (write this on your formula sheet) a 2 b 2 = (a + b)(a b) II. Perfect Square Trinomial Factoring Pattern: (write this on your formula sheet) a 2 + 2ab + b 2 = (a + b) 2 a 2 2ab + b 2 = (a b) 2 Examples: 1. Factor each polynomial. a. y 2 9 b. 64x 2 16 c. x 2 81y 2 (y + 3)(y 3) 16(2x 1)(2x + 1) (x 9y)(x + 9y) d x 2 e. x 2 + 6x + 9 f. 4n n (1 2x)(1 + 2x) (x + 3) 2 (2n + 5) 2 g. 9m 2 6my + y 2 h. 2x 2 16x 32 (3m y) 2 2(x + 4) 2 2. Solve. x x + = 0 4 (x ⁵ ₂) 2 = 0 x = ⁵ ₂

15 Notes A rock is dropped from a riverbank that is 4 feet above the surface of the river. After how many seconds does the rock hit the surface of the water? 16t = 0 4(4t 2 1) = 0 4(2t + 1)(2t 1) = 0 t = ½ second 4. A window washer drops a wet sponge from a height of 64 feet. After how many seconds does the sponge land on the ground? 16t = 0 16(t 2 4) = 0 16(t 2)(t + 2) = 0 t = 2 seconds

16 Algebra Notes Section 9.8: Factor Polynomials Completely Objective(s): To factor polynomials completely. Vocabulary : I. Guidelines for Factoring a Polynomial Completely. greatest common monomial factor. 1. Factor out the (lesson 9.4) difference of two squares 2. Look for a or a (lesson 9.7) perfect square trinomial ax 2 + bx + c 3. Factor a trinomial of the form into a product of binomial factors. grouping 4. Factor a polynomial with four terms by. Examples: 1. Factor the expression, if possible. a. 4x(x 3) + 5(x 3) b. 2y 2 (y 5) 3(5 y) (x 3)(4x + 5) 2y 2 (y 5) + 3(y 5) (y 5)(2y 2 + 3) c. x 3 + 2x 2 + 8x + 16 d. x 2 + 4x + xy + 4y (x + 2)(x 2 + 8) (x + 4)(x + y) e. x x + 2x 2 f. x 2 4x 3 (x + 2)(x 2 5) cannot be factored g. 3x 3 21x 2 54x h. 8x x 3x(x + 2)(x 9) 8x(x 2 + 3)

17 Notes Solve. a. 2x 3 18x 2 = 36x b. 3x x 2 = 24x 2x(x 3)(x 6) = 0 3x(x + 4)(x + 2) = 0 x = 0, 3, or 6 x = 0, 4, or 2 c. x 3 8x x = 0 d. x 3 25x = 0 x(x 4)(x 4) = 0 x(x + 5)(x 5) = 0 x = 0 or 4 x = 0, 5, or 5 3. A kitchen drawer has a volume of 768 in 3. The dimensions of the drawer are shown. Find the length, width, and height of the drawer. w(w + 4)(16 w) = 768 w w w 768 = 0 w 2 (w 12) + 64(w 12) = 0 (w 12)( w ) = 0 w 12 = 0 or w = 0 w = 12 or w = 8 w + 4 w 16 w w = 12 or w = 8 Dimensions could be 16 x 12 x 4 or they could be 12 x 8 x 8