Questions #10, 11, 13, 14

10. Determine whether each binomial is a factor of the given polynomial. a) x + 5, x3+ 6x2 - x - 30 b) * + 2, x4 -5x2 + 4 c) x- 2, x4- 5x2+ 6 d) 2x - 1, 2x4 - x3 - 4x2+ 2x + 1 e) 3x + 5, 3x6 + 5x5 + 9x2 + 17x - 1 f) 5x - 1, 5x4 - x3+ 10x - 10 11. The volume of a rectangular box is (x3 + 6x2 + 11x + 6) cm3 . The A box is (x + 3) cm long and (x + 2) cm wide. How high is the box? 12. a) 8x3 + 10x2 - px - 5 is divisible by 2x + 1. There is no remainder. Find the value of p. b) When x + x4 - 2x2 + k is divided by 1 + x , the remainder is 5. Find the value of k. 13. The polynomial x3 + px2 - x - 2, pER, has x - 1 as a factor. What is the value of p? 14. Let f(x) = x" - 1, where n is an integer and n 2 1. Is f(x) always divisible by x - 1? Justify your decision. 15. If the divisor of a polynomial, f(x), is x - 4, then the quotient is x'+ x - 6 and the remainder is 7. a) Write the division statement. b) Rewrite the division statement by factoring the quotient. c) Graph f(x) using your results in part b). 16. Use an example to show how synthetic division is essentially the same as regular polynomial division. Extending 17. The volume of a cylindrical can is (47x3 + 28TTx2 + 65TTx + 50TT) cm'. The can is (x + 2) cm high. What is the radius? 18. Divide. a ) ( x4+ x 3y - xy3 - 14) ) + (x2 . b ) ( x4 - 2x y + 2x2 y2 - 2xy3 + 14) = (x2+ 2) 19. Is x - y a factor of x3 - y3? Justify your answer. 20. If f ( x) = (x + 5)q(x) + (x + 3), what is the first multiple of (x + 5) that is greater than f(x)?3.5 Dividing Polynomials Module 2: Polynomial and Rational Functions MHF4U: Grade 12 University Advanced Functions Xa L.A. Chng Action Recall: Long Division Divisor 3 1 3 7 5 + Dividend Division Statement: Remember: the purpose of division is to divide a number into its factors. Apply this thinking to polynomial expressions, and we will be ablet o factor any expression. Dividing a Polynomial by a Binomial - LONG Division Ex.1 Divide 3x3 - 5x2 - 7x - 1 by x - 3 Terms of the divisor & dividend must be in descending order of degree. Dividing a Polynomial by a Binomial - SYNTHETIC Division Ex.2 Divide 3x3 - 5x2 - 7x - 1 by x - 3 Only works for binomials & must be in the form (x-k)\fEx.3 Divide 13x 2::3 + :1:4 6 byx + 2 of any missing terms Dividing o Polynomial by a TRlnomial must use LONG Division Ex.4 Divide 5x - 2x3 + 3 +x\" by 1 + 2x + x2 Using Division to Factor Polynomials How do you know whether a divisor is a factor of a dividend? Ex.5 (2x + 3) is a factor of f (x) = 6x3 + 5x2 16x 15. Determine the other factors. I A polynomial can be divided by a polynomial of the degree or LESS 0 Synthetic division is a shorter form of polynomial division. It can be used to divide a polynomial by a I When dividing polynomials: 0 Terms should be arranged in order of degree (divisor and dividend) o __ must be used as a coefficient of any missing powers of the variable 0 If the remainder of the polynomial division is zero, both the divisor and the quotient are of the dividend