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Algebra Notes Objective(s): Section 9.1: Add and Subtract Polynomials To be able to add and subtract polynomials. Recall: Coefficient (p. 97): The number part of a term with a variable part. Term of a polynomial (p. 97): The parts of an expression that are added together. Like Terms (p. 97): Terms that have the same variable parts. Constant terms are also like terms. Vocabulary : I. Monomial A number variable , whole number one or more variables with II. Degree of a Monomial: The sum , or the product of a number and exponents of the exponents. variables of the in the monomial. III. Polynomial: A a IV. Degree of a Polynomial: The V. Leading Coefficient: monomial or a term sum of monomials, each called of the polynomial. greatest degree of its terms . When a polynomial is written so that the exponents of a variable decrease from left to right, the leading coefficient is the coefficient of the first term . 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: 2x3 + x2 − 5x + 12 4 This polynomial has 2 The leading coefficient is The degree is 3 The constant term is VI. Binomial: A polynomial with 2 terms. VII. Trinomial: A polynomial with 3 terms. VIII. Adding Polynomials: To add polynomials, IX. Subtracting Polynomials: To subtract polynomials, add like terms terms. . . 12 . . add its opposite (multiply each term by −1) . Notes 9.1 Examples: 1. Consider the polynomial 3x3 − 4x4 + x2. 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. −4x4 + 3x3 + x2 e. What is the leading coefficient of the polynomial? f. List all of the coefficients of this polynomial. g. List the terms of the polynomial. −4 −4, 3, 1 −4x4, 3x3 , x2 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 Is it a polynomial? Classify by degree and number of terms 4x Yes 1st degree monomial 2x + 3x5 + 1 Yes 5th degree trinomial 1 7m 2 +m x−4 + 3 8xy + 3x2y No. Exponent must be a whole number No. Exponent cannot be negative. Yes 3rd degree binomial Notes 9.1 page 3 3. Find the sum or difference. a. (−2x2 + 3x − x3) + (3x2 + x3 − 12) x2 + 3x − 12 b. (4x3 + 2x2 − 4) + (x3 − 3x2 + x) 5x3 − x2 + x − 4 c. (2m2 − 8) − (3m2 − 4m + 1) −m2 + 4m − 9 d. (5y2 + 2y − 4) − (−y2 + 4y − 3) 6y2 − 2y − 1 4. During the period 1999 – 2005, the number of hours an individual person watched broadcast television B and cable and satellite television C can be modeled by B = 2.8t2 − 35t + 879 and C = −5t2 + 80t + 712, where t is the number of years since 1999. a. Write a polynomial that represents the total number of hours of broadcast and cable watched. B + C = −2.2t2 + 45t + 1591 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)2 + 45(3) + 1591 = 1706.2 hours Algebra Notes Objective(s): Section 9.2: Multiply Polynomials To be able to multiply polynomials. Vocabulary : I. Recall properties of multiplying and adding expressions: Examples: 2x • 4x = 8x2 2x + 4x = 6x 2x • 3x2 = 6x3 2x + 3x2 = 3x2 + 2x 2x2y3 + 4y3x2 = 6x2y3 2x3y2 + 4y3x2 = 2x3y2 + 4x2y3 2x(4x + 1) = 8x2 + 2x II. FOIL Pattern: F O (2x + 3)(4x + 1) = F O I L 2 8x + 2x + 12x + 3 = 8x2 + 14x + 3 I L Examples: 1. Find the product. a. 3x2(2x3 − x2 + 4x − 3) 6x5 − 3x4 + 12x3 − 9x2 c. (2x − 1)(3x −4) 6x2 − 11x + 4 e. (x2 − x − 2)(3x − 1) 3x3 − 4x2 − 5x + 2 b. (x + 4)(2x − 1) 2x2 + 7x − 4 d. (4x + 3)(x + 2) 4x2 + 11x + 6 Notes 9.2 2. Perform the indicated operation. a. (2x + 1) + (3x − 2) b. (2x + 1)(3x − 2) 5x − 1 6x2 − x − 2 3. A rectangle has dimensions x + 3 and x + 5. Which expression shows the area of the rectangle? A. x2 + 15 B. x2 + 3x + 15 C. x2 + 8x + 1 D. x2 + 8x E. None of these A = length x width =(x + 3)(x + 5) = x2 + 8x + 15 4. A rectangular trivet has a ceramic center and a wooden border. The dimensions of the center and border are shown in the diagram. x inches a. Write a polynomial that represents the total area of the trivet. A = length x width = (2x + 8) (2x + 6) = 4x2 + 28x + 48 8” 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 + 28(2) + 48 = 120 in2 5. Write a polynomial that represents the area of the shaded region. A = (2x - 1) (x + 2) ‒ 10 8 = 2x2 + 4x ‒ x ‒ 2 ‒ 80 = 2x2 + 3x ‒ 82 8 10 2x - 1 x+2 Algebra Notes Section 9.3: Finding Special Products of Polynomials To use special product patterns to multiply polynomials. Objective(s): Vocabulary : I. Square of a Binomial Pattern: (write this on your formula sheet) II. Sum and Difference Pattern: (write this on your formula sheet) (a + b)2 = a2 + 2ab + b2 (a − b)2 = a2 − 2ab + b2 a2 − b2 (a + b)(a − b) = Examples: 1. Find the product. a. (2x + 5)2 4x2 b. (3x − y)2 + 20x + 25 d. (4x + y)(4x − y) 16x2 − 9x2 − 6xy + e. (x + 1)(x + 1) y2 x2 + 2x + 1 2. Which special product pattern results in the following polynomial? a. x2 + 6x + 9 b. x2 − 25 (x + 3)2 (x + 5)(x − 5) c. x2 − 8x + 16 (x − 4)2 c. (x + 3)(x − 3) y2 x2 − 9 f. (2x −1)(2x − 1) 4x2 − 4x + 1 Notes 9.3 3. Use a special products pattern to find the product without a calculator: 19 • 21 19 • 21 = (20 − 1)(20 + 1) = 400 − 1 = 399 4. Use a special products pattern to find the product without a calculator: 212 212 = (20 + 1)2 = 400 + 40 + 1 = 441 5. 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 a. What percent of the possible gene combinations of the offspring result in droopy ears? E 25% e b. How can a polynomial model the possible combinations of the offspring? (0.5E + 0.5e)2 = 0.25E2 + 0.5Ee + 0.25e2 The coefficient of e2 shows that 25% of the possible gene combinations result in droopy ears. e EE Ee Straight Straight Ee ee Straight Droopy Algebra Notes Objective(s): Section 9.4: Solve Polynomial Equations in Factored Form To solve polynomial equations. Vocabulary : I. Zero-Product Property: Let a and b be real numbers. If II. Roots: The solutions to ab = 0. III. Factoring: Writing a polynomial as a product polynomials IV. Greatest Common Monomial Factor (GCF): A a • b = 0 then a = 0 or of b=0 other . monomial with an integer evenly into each of the polynomial’s terms. coefficient that divides V. Projectile: An object that is propelled into the air but has no power to keep itself in the air. VI. Vertical Motion Model: The height h (in feet) of a projectile can be modeled by the equation (write this on your formula sheet) . h = −16t2 + 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). VII. To solve a polynomial equation using the zero-product property: You may need to factor polynomials. Look for the the polynomial, or write it as a product of other GCF Examples: 1. Solve each of the following. a. (x + 3)(x − 5) = 0 −3 or 5 b. (2x + 1)(x + 4) = 0 − ½ or −4 2. Name the greatest common monomial factor of the polynomial. a. 8xy + 20x 4x b. 10x2y3 − 15xy 5xy of the polynomial's terms. Notes 9.4 3. Factor out the greatest common monomial factor. a. 8x + 12y b. 5x + 10y c. 14x2 y2 + 21y4x3 4(2x + 3y) 5(x + 2y) d. 8x3 + 10x4 + 2 e. 4x2y − 5xy + xy2 2(4x3 + 5x4 + 1) 7x2y2(2 + 3y2x) f. 27x2y3 + 18x3y2 + 9 xy(4x − 5 + y) 4. Solve. a. 3x2 + 18x = 0 3x(x + 6) = 0 x = 0 or x = −6 c. 4x2 = 14x b. 4x2 + 2x = 0 2x(2x + 1) = 0 x = 0 or x = − ½ d. 6x2 = 15x 4x2 − 14x = 0 2x(2x − 7) = 0 6x2 − 15x = 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 = − 16t2 + vt + s h = −16t2 + 32t 0 = −16t2 + 32t 0 = −16t(t − 2) t = 0 or t = 2 2 seconds 3(9x2y3 + 6x3y2 + 3) Section 9.5: Factor x2 + bx + c Algebra Notes Objective(s): To factor trinomials of the form x2 + bx + c. Vocabulary : Note: The method taught to you in this section only applies to a trinomial where the leading coefficient is 1 (ex: x2 + 5x + 6). You cannot use this method if the leading coefficient is not 1. ( ex: 4x2 + 8x + 3 ) I. Factoring x2 + bx + c: x2 + bx + c = (x + p)(x + q) provided p+q=b and p•q=c Examples: 1. Which of the following trinomials can be factored using the method of this section. Circle all that apply. A. x2 + 6x + 7 B. 6x2 + 7x + 1 D. 3x2 + 4x + 1 E. x2 − 3x − 4 C. 4x2 + 7x − 2 2. Factor each of the following. a. x2 + 11x + 18 (x + 9)(x + 2) d. x2 − 6x + 8 (x − 2)(x − 4) g. x2 + 3x − 10 (x + 5)(x − 2) b. x2 + 5x + 6 (x + 3)(x + 2) e. x2 + 2x − 15 (x + 5)(x − 3) c. x2 − 9x + 20 (x − 5)(x − 4) f. x2 − 5x + 6 (x − 2)(x ‒ 3) . Notes 9.5 3. Factor each of the following. a. x2 + 5x + 6 b. x2 − x − 6 (x + 3)(x + 2) (x − 3)(x + 2) c. x2 + x − 6 d. x2 − 5x + 6 (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. x2 + 3x = 18 b. x2 − 2x = 24 (x +6)(x − 3) = 0 (x − 6)(x + 4) = 0 x = −6 or x = 3 x = 6 or x = −4 c. x2 = 3x + 28 (x − 7)(x + 4) = 0 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 + 4) = 117 w2 + 4w − 117 = 0 w 2" w+2 (w + 13)(w − 9) = 0 w = −13 or w = 9. w = −13 does not make sense with the problem. Therefore, the width must be 9”. Section 9.6: Factor ax2 + bx + c Algebra Notes Objective(s): To factor trinomials of the form x2 + bx + c. Vocabulary : I. Two methods for factoring ax2 + bx + c: 1. Guess and check with factors of a and c: Factor 2x2 − 7x + 3 Factors of 2 Factors of 3 1, 2 −1, −3 (x − 1)(2x − 3) −3x − 2x = −5x 1, 2 −3, −1 (x − 3)(2x − 1) −x − 6x = − 7x Answer: Possible Factorization Middle term multiplied (x − 3)(2x − 1) 2. Grouping method: Factor 3x2 + 14x − 5 Step 1: Find two numbers whose product is: and whose sum is: ac b product must be (−5)(3) = −15 and the sum must be 14 15 and −1 work. 15(−1) = −15 and 15 + −1 = 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. 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. 3x2 + 14x − 5 3x2 + 15x − x − 5 3x(x + 5) − 1(x + 5) (x + 5)(3x − 1) II. Factoring when a is negative: To factor a trinomial of the form ax2 + 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. Notes 9.6 Examples: 1. Factor any two of the following by guess and check and factor the other two by grouping. a. 2x2 − 13x + 6 (x − 6)(2x − 1) b. 4x2 − 12x − 7 c. 3x2 + 8x + 4 d. 4x2 − 9x + 5 (2x + 1)(2x − 7) (x + 2)(3x + 2) (x − 1)(4x − 5) 2. Factor each of the following using any method. a. −4x2 + 12x + 7 −(2x +1)(2x − 7) b. −3x2 − x + 2 c. −3x2 − 13x + 4 −(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. b. After how many seconds does it hit the ground? h = − 16t2 + vt + s h = − 16t2 + 28t + 8 0 = −4(4t2 − 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 4w2 + 5w − 6 = 0 (4w − 3)(w + 2) = 0 w = ¾ or w = −2 w = ¾ ft Algebra Notes Objective(s): Section 9.7: Factor Special Products To factor special products. Vocabulary : I. Difference of Squares Factoring Pattern: (write this on your formula sheet) a2 − b 2 = (a + b)(a − b) II. Perfect Square Trinomial Factoring Pattern: (write this on your formula sheet) a2 + 2ab + b2 = a2 − 2ab + b2 = (a + b)2 (a − b)2 Examples: 1. Factor each polynomial. a. y2 − 9 b. 64x2 − 16 (y + 3)(y − 3) d. 12 − 48x2 16(2x − 1)(2x + 1) e. x2 + 6x + 9 12(1 − 2x)(1 + 2x) (x + 3)2 g. 9m2 − 6my + y2 h. −2x2 − 16x − 32 (3m − y)2 −2(x + 4)2 2. Solve. x2 − 5x + 25 =0 4 (x − ⁵⁄₂)2 = 0 x = ⁵⁄₂ c. x2 − 81y2 (x − 9y)(x + 9y) f. 4n2 + 20n + 25 (2n + 5)2 Notes 9.7 3. 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? −16t2 + 4 = 0 −4(4t2 − 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? −16t2 + 64 = 0 −16(t2 − 4) = 0 −16(t − 2)(t + 2) = 0 t = 2 seconds Algebra Notes Objective(s): Section 9.8: Factor Polynomials Completely To factor polynomials completely. Vocabulary : I. Guidelines for Factoring a Polynomial Completely. 1. Factor out the 2. Look for a (lesson 9.7) greatest common monomial factor. difference of two squares 3. Factor a trinomial of the form factors or a (lesson 9.4) perfect square trinomial ax2 + bx + c into a product of grouping . binomial . 4. Factor a polynomial with four terms by Examples: 1. Factor the expression, if possible. a. 4x(x − 3) + 5(x − 3) (x − 3)(4x + 5) b. 2y2(y − 5) − 3(5 − y) 2y2 (y − 5) + 3(y − 5) (y − 5)(2y2 + 3) c. x3 + 2x2 + 8x + 16 (x + 2)(x2 + 8) e. x3 − 10 − 5x + 2x2 (x + 2)(x2 − 5) g. 3x3 − 21x2 − 54x 3x(x + 2)(x − 9) d. x2 + 4x + xy + 4y (x + 4)(x + y) f. x2 − 4x − 3 cannot be factored h. 8x3 + 24x 8x(x2 + 3) Notes 9.8 2. Solve. a. 2x3 − 18x2 = − 36x b. 3x3 + 18x2 = −24x 2x(x − 3)(x − 6) = 0 3x(x + 4)(x + 2) = 0 x = 0, 3, or 6 x = 0, −4, or −2 c. x3 − 8x2 + 16x = 0 d. x3 − 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 in3. The dimensions of the drawer are shown. Find the length, width, and height of the drawer. w(w + 4)(16 − w) = 768 −w3 + 12w2 + 64w − 768 = 0 −w2(w − 12) + 64(w − 12) = 0 (w − 12)(−w2 + 64) = 0 16 − w w w+4 w − 12 = 0 or −w2 + 64 = 0 w = 12 or w = 8 w = 12 or w = 8 Dimensions could be or they could be 16 x 12 x 4 12 x 8 x 8