Worksheet 4.7 Polynomials

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Solution a) By trial and error notice that p(2) = 48 − 66 + 22 − 2 = 0 i.e. 2 is a root of p(x). So x − 2 is a factor of p(x). To find other factors we’ll divide p(x) by (x − 2). 6x 2 − 5x + 1 x − 2 6x 3 − 17x 2 + 11x − 2 6x 3 − 12x 2 −5x 2 + 11x −5x 2 + 10x x − 2 x − 2 0 So p(x) = (x − 2)(6x 2 − 5x + 1). Now notice 6x 2 − 5x + 1 = 6x 2 − 3x − 2x + 1 = 3x(2x − 1) − (2x − 1) = (3x − 1)(2x − 1) So p(x) = (x − 2)(3x − 1)(2x − 1) and its factors are (x − 2), (3x − 1) and (2x − 1). Solution b) The solutions to p(x) = 0 occur when That is, Exercises: x − 2 = 0, 3x − 1 = 0, 2x − 1 = 0. x = 2, x = 1 1 , x = 3 2 . 1. For each of the following polynomials find (i) its factors; (ii) its roots. (a) x 3 − 3x 2 + 5x − 6 (b) x 3 + 3x 2 − 9x + 5 (c) 6x 3 − x 2 − 2x (d) 4x 3 − 7x 2 − 14x − 3 2. Given that x − 2 is a factor of the polynomial x 3 − kx 2 − 24x + 28, find k and the roots of this polynomial. 6
3. Find the quadratic whose roots are −1 and 1 3 and whose value at x = 2 is 10. 4. Find the polynomial of degree 3 which has a root at −1, a double root at 3 and whose value at x = 2 is 12. 5. (a) Explain why the polynomial p(x) = 3x 2 + 11x 2 + 8x − 4 has at least 1 root in the interval from x = 0 to x = 1. (b) Find all the roots of this polynomial. 7
Page 1 and 2: A polynomial is an expression of th
Page 3 and 4: Exercises: So 3x 3 − 2x 2 + 4x +
Page 5: Note: Checking this using long divi
Page 9 and 10: Answers 4.7 Section 1 1. (a) (i) 1
Page 11: 5. (a) k = 11, factors are (x − 4

Worksheet 4.7 Polynomials

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