Multiplication of Polynomials

Transcription

1 Multiplication of Polynomials In multiplying polynomials, we need to consider the following cases: Case 1: Monomial times Polynomial In this case, you can use the distributive property and laws of exponents (Product of Powers) to multiply a polynomial by a monomial a a 3a 5 6 1a 6 3a a 18a 30 Simplify x y x 4x y 3xy 5 x y 54 x y 5 3 x y Distributive Property and Product of Powers Distributive Property and Product of Powers x y 0x y 15x y Simplify. m m m 3. n 8n 3n 7 = m m m 8 n 3 n m 7n m = m 3m 16n 6n 14n m Distributive Property and Product of Powers Simplify. Case : Binomial Times Binomial Case can be solved using 3 methods. a. Distributive Property When multiplying two binomials, use the distributive property twice. That is, (a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd Then, simplify by combining similar terms. 1. (4a + 7)(a 5) Use distributive property and then combine similar terms. 4a(a 5) + 7( a 5) = 8a 0a + 14a 35 = 8a 6a 35. 6x y3x 5y = 6x(3x 5y) y(3x 5y) = 18x 30xy 3xy + 5y Distributive Property = 18x 33xy + 5y Simplify

2 3. (x y + 7y 3 )(3x y + 4y 3 ) = x y(3x y + 4y 3 ) + 7y 3 (3x y + 4y 3 ) Distributive Property = 6x 4 y + 8x y 4 + 1x y 4 + 8y 6 Simplify a b a b a a b b a b 4. x y x y x x y y x y = x a a b a b b x y x y y Distributive Property = x a + x a y b + y b Simplify b. FOIL Method A shortcut of distributive property is called the FOIL method. It is an acronym and tells you which terms to multiply. The letters FOIL stand for First, Outer, Inner and Last. First means multiply the terms which occur in each binomial. Outer means multiply the innermost two terms. Inner means multiply the innermost terms. Last means multiply the terms which occur last in each binomial. Make sure you combine any like terms Use FOIL method to find the product of the two binomials. 1. (x + 5)(x + 3) F L (x + 5)(x + 3) I O (x + 5)(x + 3) = (x)(x) + (3)(x) + ()(x) + (3)(5) = x + 3x + x + 15 = x + 5x (4x + 3y)(x 5y) = (4x)(x) + (4x)(- 5y) + (x)(3y) + (3y)(-5y) = 6x 0xy + 6xy 15y = 6x 14xy 15y 3. (x 3y)(3x + 5y) = (x )(3x ) + (x )(5y) + (3x )(- 3y) + (-3y)(5y) = 6x x y 3x y 15y 4. (7x a 3y b )(x a + y b ) = (7x a )(x a ) + (7x a )(y b ) + (- 3y b )(x a ) + (-3y b )(y b ) = 7x a + 4x a y b 3y b

3 c. Vertical Method If you want to multiply 45 and 3, you would probably not want to try to multiply them horizontally. It is easy to multiply them vertically. 45 x You can multiply two binomials in this same manner. Study the examples below. 1. (8x + 3)(x + 5) 8x + 3 x x x (8x + 3) + 16x + 6x x(8x + 3) 16x + 46x (6x 5y)(x + y) 6x 5y x x + y 6xy 5y y(6x 5y) + 1x 10xy x(6x 5y) 1x 4xy 5 Case 3: a. Binomial Times Trinomial b. Trinomial Times Trinomial c. Multinomial Times Multinomial In this case, FOIL method is not applicable. We can use either the distributive property method or the vertical method. 1. Use distributive to find the product of (a 3b)(a + ab + 1). (a 3b)(a + ab + b ) = a(a + ab + b ) 3b(a + ab + b ) = a 3 + 4a b + ab 3a b 6ab 3b 3 Distributive Property

4 = a 3 + (4a b 3ab ) + (ab 6ab ) 3b 3 Collect similar terms. = a 3 + a b 4ab 3b Simplify.. Find the product of 4x 3x + and x + x 1 using vertical method. 4x 3x + x x + x 1 4x + 3x 1 (4x 3x+ ) 4x 3 3x + x x(4x 3x+ ) + 4x 4 3x 3 + x x (4x 3x+ ) 4x 4 + x 3 5x + 5x 3. Multiply a 4 + 3a 3 5a + a 1 and a 4 a 3 + a 3a +. In you are multiplying two multinomials, it is better to use the vertical method. a 4 + 3a 3 5a + a 1 x a 4 a 3 + a 3a + a 4 + 6a 3 10a + a - (a 4 + 3a 3 5a + a 1) 3a 5 9a a 3 3a 3a - 3a(a 4 + 3a 3 5a + a 1) a 6 + 3a 5 5a 4 + a 3 a a (a 4 + 3a 3 5a + a 1) a 7 6a a 5 a 4 + a 3 - a 3 (a 4 + 3a 3 5a + a 1) + a 8 + 3a 7 5a 6 + a 5 a 4 a 4 (a 4 + 3a 3 5a + a 1) a 8 + a 7 11a a 5 15a 4 + 4a 3 14a a TRY THESE! A. Find the product using distributive property. 1. (3x 9x + 11) 6. 10a(a 3 + a 5a + 6). 5n (5n 3 9n + n ) 7. 3x 3 y z 6 (x 5 y 7 z + 5x 4 y 7 z 10 x y z) 3. a b(7a 3 b + a b 3ab 3 + ) 8. 1/ ( 1/ + 5/ 7/ ) 4. x y (4x xy + 7y ) 9. x 4/5 (3x 1/ 5x 1/4 + 7x 1/3 ) 5. X a y a+ (x 3a + 5 y a x 7a y a + 7 ) 10. 4a b 3 (8 a 5 b 3 + 6a 7 b 4 ) B. Find the product of the two binomials using FOIL method. 1. (x 5)(x 1) 6. (x 3)(x 7). (a + 7)(a + ) 7. (6y 3)(y + 4) 3. (r 5)(r + 10) 8. (1a + 7)(3a 1) 4. (t 1)(t 7) 9. (8k 3)(k 11) 5x 5 x 1 5. (p + 9)(p + 3)

5 C. Find the product using the vertical method. 1. (m 1)(m 4) 6. (7x + y )(x 5y ). (n + 8)(n + 10) 7. (3a 3 1)(a 3 + 5) 3. (a + 6)(a 1) 8. (x 3/4 + )(x /5 +1) 4. (p + 11)(p ) 9. (n 1/ + 3n 1/3 )(n 1/ 5n 1/) 5. (t 1)(t + 15) 10. (6xy 5z)(xy + 3z) LETS US DO THIS A. Simplify the following. 1. (w 3)(w -10) 6. n(n + 1) 3 + x(n + 1). 5x(x + 3)(x ) 7. (x + 1)(x + )(x + 3) 3. (a + 7)(a 1) (1 8a) 8. (y + )(y 3 + y y + 1) 4. (t 5) + (t 1) 9. (1 r)( 5r + 3r ) 5. 3(p + )(p 1) + (p 3) 10. 3(k 1)(k + 1) (k + 3)(k 3) + (k 1) B. Solve the following. 1. Subtract the product of 3x + 7 and x 1 from 5x 8x Add the product of a + 5 and a 5 to the square of a What must be multiplied to 8n + 3 to get 64n 9? 4. From the sum of y + y 5 and y + 3y +, subtract the product of y + 3 and y. 5. Find a + b + c if 4(3x x + ) (x + 3x 1) = ax + bx + c. EXPRESS YOUR IDEAS 1. What is the difference between the equations (n + 4) = n + 8n + 16 and (n + 4) = n + 16?. The product of two binomials is always a trinomial. Do you agree with this statement? 3. Among the three methods of multiplying two binomials, which is the easiest? Depend your answer. 4. Explain how would you perform the multiplication: - 3x(x + )(x + 5) 5. What does FOIL stand for? What is the objective of FOIL?

6 STRIVE HARDER 1 x? 1 1. If x + 10, what is x x. Expand the following. a. (a + 3b + c) b. (x y + z) c. (3m n + 4m) Make a conjecture about the expansion of (a + b + c). y x 3. If 3and x + y = 3, find the value of x y. 4. Multiply and give the pattern. a. (x + 5)(x 5) = b. (n + 9)(n 9) = c. (a + 11)(a 11) = d. (y 4)(y + 4) = e. (k 6)(k + 6) = Pattern: f. (t +8) = g. (p ) = h. (x + 7) = i. (h 3) = j. (w + 10) = Pattern: k. (x + 3) 3 = l. (c + 1) 3 = m. (d + 5) 3 = n. (z + ) 3 = o. (b + 3) 3 = Pattern: p. (a ) 3 = q. (x 1) 3 = r. (n 7) = s. (y 3) 3 =

7 CLICK I.T. 1. Use the website (multiplying binomials calculator) to calculate the product of the two binomials (ax + b). a. (8x + 7)(5x 1) f. (7x 1)(18x + 7) b. (x 11)(x + 1) g. (50x + 11)(13x 1) c. (15x )(3x 19) h. (.67x 1.33)(7.63x 8.1) d. (11x + 33)(4x 9) i. (1.79x +4.33)(1.3x 1.30) e. (3x 9)(8x + 3 j. (0.83x 0.3)(0.65x 1.78)