3.1 Polynomials MATHPOWER TM 10, Ontario Edition, pp. 128 133 To add polynomials, collect like terms. To subtract a polynomial, add its opposite. To multiply monomials, multiply the numerical coefficients. Then, multiply the variables using the exponent rules for multiplication. To divide monomials, divide the numerical coefficients. Then, divide the variables using the exponent rules for division. To multiply a polynomial by a monomial, use the distributive property to multiply each term in the polynomial by the monomial. 1. Classify each polynomial by degree and by number of terms. a) 3x 2 2x b) 4a 2 b 3 5. Simplify. a) (6x)(2x 2 ) b) (5pq 2 )( 4p 2 q 2 ) c) (3ab)( 2ab 2 )(2a 3 ) d) ( 6x 2 yz)( 5y 3 z) c) 8 + 2y 4 + 3y 3 d) 4x 5 2x 3 + x 2 + 4 2. Evaluate each expression for the given value(s) of the variable(s). a) 5x 2 4x + 9 for x = 2 e) 6 15x 2 5x f) g) 2 2 21xyz 2 7xy z h) 3 2 24ab 2 3ab 32pq 2 3 8pq 2 4 b) 2x 2 4xy 5y 2 for x = 3, y = 2 3. Write each polynomial in descending order of x. a) 6 + 4x 2 5x 5 + 3x 2x 2 b) 3x 2 y 4 + 4x 4 y 2 x 3 y 3 + x 5 y 2xy 5 4. Simplify. a) (6y 2) + (2y + 8) b) (a + 2b) + (3a 4b) 6. Communication Explain how to simplify and evaluate 3x(x + 1) 4(x 2 3x) for x = 2. 7. Expand and simplify. a) 2z + 3(4z 2) + 2(4 3z) b) 3x(x 2 + 2x 2) + 2x(3x 2 x 4) c) 4m(m 2 mn n 2 ) 2n(6m 2 + mn + 4n 2 ) c) (8 + 6x) (9 + x) d) (x + y) (x y) e) (3x 2 + 2x 6) + (2x 2 4x + 7) 8. Application Write a polynomial with three terms and degree 4. f) (5a 2 b + 2ab 3b 2 ) (6a 2 b 3ab + b 2 ) g) (3y 2 y 6) (2y 2 + 5y 7) 9. Problem Solving For the rectangular prism, write an expression that represents a) the volume b) the surface area 4y 3x 2y Chapter 3 29
3.2 Multiplying Binomials MATHPOWER TM 10, Ontario Edition, pp. 134 139 To find the product of two binomials, use either of the following. a) the distributive property b) FOIL, which stands for the sum of the products of the First terms, Outside terms, Inside terms, and Last terms Verify the product of two binomials by substituting a convenient value for the variable in the original product and in the simplified expression. 1. Communication Explain how the diagram models the product. x + y 2x 2x 2 2xy + 3 3x 3y 3. Expand and simplify. a) 2(m 3)(m + 8) b) 3(x + 2)(x + 3) 2. Find the product. a) (a + 3)(a + 2) c) 2(y 3)(y + 2) d) 0.2(x + 1)(x + 2) b) (2 + k)(3 + k) e) 3(6x 2y)(2x 3y) c) (c 5)(c 3) d) (t + 5)(t 1) f) (x + 3)(x + 2) + (x + 4)(x + 1) e) (3 b)(4 + b) g) (y 4)(y 3) (y 2)(y + 5) f) (6v + 3)(v + 1) h) (3w 2)(w + 4) + (2w + 3)(4w 1) g) (5 + 2x)(2 + x) i) 6(m 2)(m + 3) 3(3m 4) h) (y 5)(2y 2) j) 4(2x + 3)(2x + 3) 10 + 3(3x 1)(3x 1) i) (m + 4)(3m 2) Problem Solving j) (4g 3)(g + 4) k) (2y + 3)(3y + 2) 4. Write and simplify an expression to represent the area of the figure. x 4 x + 6 x 1 l) (5h 1)(2h 3) m) (3 2s)(2 3s) n) (4 + 2p)( 3 4p) o) ( 2t r)( 3t + r) 5. Write and simplify an expression to represent the area of the shaded region. 2y + 1 2y x 2y 2y + x 30 Chapter 3
3.3 Special Products MATHPOWER TM 10, Ontario Edition, pp. 140 145 To square a binomial, use one of the following patterns. (a + b) 2 = a 2 + 2ab + b 2 (a b) 2 = a 2 2ab + b 2 To find the product of the sum and difference of two terms, use the following pattern. (a + b)(a b) = a 2 b 2 1. Expand. a) (x + 4) 2 b) (y 7) 2 c) (m 2)(m + 2) d) (x 5)(x + 5) e) 2(6x 3) 2 f) 3(5 + 4t) 2 g) (3y 3)(3y + 3) h) (5m + 2n)(5m 2n) 3. Application Complete the table. Numbers (a + b)(a b) Product a) 25 15 (20 + 5)(20 5) b) (30 + 6)(30 6) c) 27 33 d) (20 4)(20 + 4) 4. Expand and simplify. a) (x 2 + 2) 2 i) (3x + 4y) 2 j) 2(a 7b) 2 b) (2y 2 3) 2 2. Expand and simplify. a) (m 6) 2 (m + 2)(m 2) c) (y 2 + 3)(y 2 3) b) (x + 4)(x 3) 3(x + 2) 2 d) (4m 2 + n 2 )(4m 2 n 2 ) c) 3(2b 1) 2 2(4b 5) 2 e) ( 3x 5)( 3x + 5) + (x + 1) 2 d) (x + 5)(x 5) + (3x 1)(3x + 1) e) 4x 2 (2 3x) 2 + 6(2x 1)(2x + 1) 5. Problem Solving The length of an edge of a cube is represented by the expression 3x 2y. a) Write, expand, and simplify an expression for the surface area of the cube. f) (2a 1)(2a + 1) (a 3)(a + 3) b) If x represents 4 cm and y represents 3 cm, calculate the surface area, in square centimetres. Chapter 3 31
3.4 Common Factors MATHPOWER TM 10, Ontario Edition, pp. 147 152 To factor a polynomial with a common monomial factor, remove the greatest common factor of the coefficients and the greatest common factor of the variable parts. To factor a polynomial with a common binomial factor, think of the binomial as one factor. To factor a polynomial by grouping, group pairs of terms with a common factor. 1. Factor, if possible. a) 4x + 28 b) 3x + 17 3. Factor by grouping. a) ax by + xb ya c) 6x 32y d) 26x 2 13y b) y 2 x + y xy e) 2ax + 10ay 8az f) 2a 2 6a 15 c) ab + 9 + 3a + 3b g) 8x 2 + 32y 3 h) 10y 5y 2 + 25y 3 d) t 2 tr + 4r 4t i) 14rst + 7rs 6t j) 36xy 12x 2 y e) 4x 2 + 6xy + 12y + 8x k) 4ab 2 + 2a 2 c + 5b 2 c 2 f) 3x 2 y 6x 2 2y + y 2 l) 3x 3 y 2 12x 2 y 3 + 18x 2 y + 15xy 2 g) 4ab 2 12a 2 b 3bc + 9ac 2. Factor, if possible. a) 3x(y z) 2(y z) b) 5y(z + 3) + x(z 3) 4. Problem Solving Write an expression for the area of each shaded region in factored form. a) 3x c) 4t(r + 6) (r + 6) d) 7(a + b) 2x(a + b) b) 2y y + 1 3x 3y 4x 2 e) 2x(3m 5) 3(5 3m) c) 4r 32 Chapter 3
3.5 Factoring x 2 + bx + c MATHPOWER TM 10, Ontario Edition, pp. 153 158 To factor a trinomial in the form x 2 + bx + c, a) write x as the first term in each binomial factor b) write the second terms, which are two numbers whose sum is b and whose product is c When factoring a trinomial, first remove any common factors. 1. Factor, if possible. a) x 2 5x + 6 3. Factor completely. a) 2x 2 + 10x + 12 b) y 2 + 2y 3 b) 3x 2 + 9x 12 c) m 2 + 7m 12 c) 5x 2 35x + 50 d) a 2 + 6a + 5 d) 4x 2 16x 48 e) x 2 9x 10 e) 2x 2 16x 66 f) b 2 7b + 10 f) x 3 13x 2 + 42x g) y 2 6y + 7 h) x 2 + x 20 2. Factor, if possible. a) x 2 + 24x 52 4. Application The area of a doubles tennis court can be represented approximately by the trinomial x 2 x 42. a) Factor x 2 x 42 to find binomials that represent the length and width of a doubles tennis court. b) m 2 18m + 45 c) x 2 + 5x 36 b) If x represents 17.8 m, find the length and width of a doubles tennis court, to the nearest tenth of a metre. d) x 2 5xy 66y 2 e) m 2 + 12mn + 32n 2 f) 42 + y y 2 5. Communication Find two values for k such that the trinomial can be factored over the integers. Explain your reasoning. a) x 2 9x + k g) 32 + 4x x 2 b) x 2 kx + 6 h) x 4 + 7x 2 + 12 Chapter 3 33
3.6 Factoring ax 2 + bx + c, a 1 MATHPOWER TM 10, Ontario Edition, pp. 159 164 To factor a trinomial in the form ax 2 + bx + c, either use guess and check or break up the middle term. To factor by guess and check, list all the possible pairs of factors and expand to see which pair gives the correct middle term of the trinomial. To factor by breaking up the middle term, a) replace the middle term, bx, by two terms whose coefficients have a sum of b and a product of a c b) group pairs of terms and remove a common factor from each pair c) remove the common binomial factor 1. Factor, if possible. a) 3y 2 + y 4 b) 3y 2 + 5y + 1 f) 6x 2xy 8y 2 c) 2a 2 13a + 21 d) 4n 2 + 7n 5 g) 6m 2 13mn 5n 2 e) 20x 2 7x 6 f) 18y 2 + 15y 18 h) 9x 2 + 3xy 20y 2 g) 5x 2 12x 6 h) 8m 2 + 6m 20 i) 12a 2 + 28ab 24b 2 2. Factor. a) 2x 2 + 5xy 2y 2 3. Communication Describe how to factor 18a 2 21ab + 6b 2. b) 3y 2 + 2yz z 2 c) 15x 2 13xy + 2y 2 4. Application The area of a rectangular lot in a new housing development can be represented approximately by the trinomial 12x 2 + 8x 15. a) Factor the expression 12x 2 + 8x 15 to find binomials that represent the length and width of the lot. d) 6m 2 + 7mn + n 2 e) 4a 2 9ab 9b 2 b) If x represents 21 m, what are the length and width of the lot, in metres? 34 Chapter 3
3.7 Factoring Special Quadratics MATHPOWER TM 10, Ontario Edition, pp. 165 170 To factor a polynomial in the form a 2 b 2, use the pattern for the difference of squares. a 2 b 2 = (a + b)(a b) To factor a perfect square trinomial, use the patterns for squaring binomials. a 2 + 2ab + b 2 = (a + b) 2 a 2 2ab + b 2 = (a b) 2 1. Factor, if possible. a) x 2 25 b) y 2 49 3. Factor fully, if possible. a) x 2 196 b) 36y 2 + 6y + 1 c) y 4 1 d) z 2 + 64 c) 16a 4 + 40a + 25 d) 4x 2 36 e) 4a 2 9 f) 49 64m 2 e) y 2 + 100 f) p 2 4pq + 4q 2 g) 169a 2 b 2 h) 24 + 4x 2 g) 36x 2 81y 2 h) m 3 25m i) 5n 3 30n 2 + 45n j) 64x 2 16 i) 81x 2 121p 2 j) 49 (a z) 2 k) 4b 2 + 121 l) x 4 13x 2 + 36 2. State whether each trinomial is a perfect square trinomial. If it is, factor it. a) x 2 + 8x + 16 b) y 2 14y + 49 c) z 2 9z + 9 d) 9t 2 + 6t + 1 Applications 4. Evaluate each difference of squares by factoring. a) 38 2 32 2 b) 55 2 45 2 e) 4m 2 12m 9 f) 4x 2 20x + 25 c) 760 2 240 2 g) 121 22m + m 2 h) 16x 2 + 24xy + 9y 2 i) 64a 2 30ab + 49b 2 5. Determine the value(s) of k such that each trinomial is a perfect square. a) x 2 + kx + 49 b) 9x 2 + kx + 25 c) 4x 2 12x + k d) kx 2 40xy + 16y 2 Chapter 3 35
Answers CHAPTER 3 Polynomials 3.1 Polynomials 1. a) degree 2, binomial b) degree 5, monomial c) degree 4, trinomial d) degree 5, polynomial of 4 terms 2. a) 21 b) 22 3. a) 5x 5 + 4x 3 2x 2 + 3x + 6 b) x 5 y + 4x 4 y 2 x 3 y 3 + 3x 2 y 4 2xy 5 4. a) 8y + 6 b) 4a 2b c) 5x 1 d) 2y e) 6x 2 2x + 1 f) a 2 b + 5ab 4b 2 g) y 2 6y + 1 5. a) 12x 3 b) 20p 3 q 4 c) 12a 5 b 3 d) 30x 2 y 4 z 2 e) 3x 4 f) 8ab g) 3x h) 4q 6. Multiply 3x(x + 1) to get 3x 2 + 3x. Then, multiply 4(x 2 3x) to get 4x 2 + 12x. Then, collect the like terms to get x 2 + 15x. Then, substitute 2 for each x and evaluate to get (2) 2 + 15(2) = 4 + 30 = 26. 7. a) 8z + 2 b) 9x 3 + 4x 2 14x c) 4m 3 16m 2 n 6mn 2 8n 3 8. Answers may vary. 2m 3 n + 3m 8n 9. a) 24xy 2 b) 36xy + 16y 2 3.2 Multiplying Binomials 1. The length of the rectangle is 2x + 3. The width is x + y. The area is (2x + 3)(x + y) = 2x 2 + 3x + 2xy + 3y. 2. a) a 2 + 5a + 6 b) 6 + 5k + k 2 c) c 2 8c + 15 d) t 2 + 4t 5 e) 12 b b 2 f) 6v 2 + 9v + 3 g) 10 + 9x + 2x 2 h) 2y 2 12y + 10 i) 3m 2 + 10m 8 j) 4g 2 + 13g 12 k) 6y 2 13y + 6 l) 10h 2 17h + 3 m) 6 13s + 6s 2 n) 12 22p 8p 2 o) 6t 2 + rt r 2 3. a) 2m 2 + 10m 48 b) 3x 2 + 15x + 18 c) 2y 2 + 2y + 12 d) 0.2x 2 + 0.6x + 0.4 e) 36x 2 66xy + 18y 2 f) 2x 2 + 10x + 10 g) 10y + 22 h) 11w 2 + 20w 11 i) 6m 2 3m 24 j) 43x 2 + 30x + 29 4. 4x + (x + 2)(x 1) = x 2 + 5x 2 or x(x + 6) (1)(x + 2) = x 2 + 5x 2 5. (2y + 1)(2y + x) (2y x)(2y) = 4xy + 2y + x 3.3 Special Products 1. a) x 2 + 8x + 16 b) y 2 14y + 49 c) m 2 4 d) x 2 25 e) 72x 2 72x + 18 f) 75 + 120t + 48t 2 g) 9y 2 9 h) 25m 2 4n 2 i) 9x 2 + 24xy + 16y 2 j) 2a 2 28ab + 98b 2 2. a) 12m + 40 b) 2x 2 11x 24 c) 20b 2 + 68b 47 d) 10x 2 26 e) 19x 2 + 12x 10 f) 3a 2 + 8 3. Numbers (a + b)(a b) Product a) 25 15 (20 + 5)(20 5) 375 b) 36 24 (30 + 6)(30 6) 864 c) 27 33 (30 3)(30 + 3) 891 d) 16 24 (20 4)(20 + 4) 384 4. a) x 4 + 4x 2 + 4 b) 4y 4 12y 2 + 9 c) y 4 9 d) 16m 4 n 4 e) 10x 2 + 2x 24 5. a) 6(3x 2y) 2 = 54x 2 72xy + 24y 2 b) 216 cm 2 3.4 Common Factors 1. a) 4(x + 7) b) does not factor c) 2(3x 16y) d) 13(2x 2 y) e) 2a(x + 5y 4z) f) does not factor g) 8(x 2 + 4y 3 ) h) 5y(2 y + 5y 2 ) i) does not factor j) 12xy(3 x) k) does not factor l) 3xy(x 2 y 4xy 2 + 6x 5y) 2. a) (3x 2)(y z) b) does not factor c) (4t 1)(r + 6) d) (7 2x)(a + b) e) (2x + 3)(3m 5) 3. a) (x y)(a + b) b) (y x)(y + 1) c) (a + 3)(b + 3) d) (t 4)(t r) e) 2(x + 2)(2x + 3y) f) (3x 2 + y)(y 2) g) (4ab 3c)(b 3a) 4. a) 3x(3πx 2y) b) 2x 2 (5y 1) c) 16r 2 (π 2) Chapter 3 37
3.5 Factoring x 2 + bx + c 1. a) (x 3)(x 2) b) (y + 3)(y 1) c) does not factor d) (a + 5)(a + 1) e) (x 10)(x + 1) f) (b 5)(b 2) g) does not factor h) (x + 5)(x 4) 2. a) (x + 26)(x 2) b) (m 15)(m 3) c) (x + 9)(x 4) d) (x 11y)(x + 6y) e) (m + 4n)(m + 8n) f) (6 + y)(7 y) g) (8 x)(4 + x) h) (x 2 + 4)(x 2 + 3) 3. a) 2(x + 2)(x + 3) b) 3(x + 4)(x 1) c) 5(x 5)(x 2) d) 4(x 6)(x + 2) e) 2(x 11)(x + 3) f) x(x 6)(x 7) 4. a) (x 7)(x + 6) b) 10.8 m by 23.8 m 5. Answers may vary. a) k = 20 because two factors with the sum of 9 are 5 and 4. x 2 9x + 20 = (x 5)(x 4); k = 14 because two factors with the sum of 9 are 7 and 2. x 2 9x + 14 = (x 7)(x 2) b) k = 7 because two factors of 6 are 1 and 6 and their sum is 7. x 2 + 7x + 6 = (x + 1)(x + 6); k = 5 because two factors of 6 are 3 and 2 and their sum is 5. x 2 + 5x + 6 = (x + 3)(x + 2) 3.7 Factoring Special Quadratics 1. a) (x + 5)(x 5) b) (y + 7)(y 7) c) (y 2 + 1)(y + 1)(y 1) d) does not factor e) (2a + 3)(2a 3) f) (7 + 8m)(7 8m) g) (13a + b)(13a b) h) does not factor i) (9x + 11p)(9x 11p) j) (7 + a z)(7 a + z) 2. a) yes, (x + 4) 2 b) yes, (y 7) 2 c) no d) yes, (3t + 1) 2 e) no f) yes, (2x 5) 2 g) yes, (11 m) 2 h) yes, (4x + 3y) 2 i) no 3. a) (x + 14)(x 14) b) does not factor c) (4a + 5) 2 d) 4(x + 3)(x 3) e) does not factor f) (p 2q) 2 g) 9(2x + 3y)(2x 3y) h) m(m 5)(m + 5) i) 5n(n 3) 2 j) 16(2x 1)(2x + 1) k) does not factor l) (x + 3)(x 3)(x + 2)(x 2) 4. a) 420 b) 1000 c) 520 000 5. a) ±14 b) ±30 c) 9 d) 25 3.6 Factoring ax 2 + bx + c, a 1 1. a) (3y + 4)(y 1) b) does not factor c) (2a 7)(a 3) d) does not factor e) (4x 3)(5x + 2) f) 3(2y + 3)(3y 2) g) does not factor h) 2(m + 2)(4m 5) 2. a) (x + 2y)(2x + y) b) (3y z)(y + z) c) (5x y)(3x 2y) d) (6m + n)(m + n) e) (4a + 3b)(a 3b) f) 2(x + y)(3x 4y) g) (2m 5n)(3m + n) h) (3x + 5y)(3x 4y) i) 4(3a 2b)(a + 3b) 3. Remove the common factor to get 3(6a 2 7ab +2b 2 ). Then, factor the trinomial by guess and test to get 3(3a 2b)(2a b). 4. a) (2x + 3)(6x 5) b) 45 m by 121 m 38 Chapter 3