Topic 1 .2 - 1.3 Polynomials

Topic 1.2: Polynomials, quotient and remainder You should be able to:  add, subtract, multiply and divide polynomials  understand the words ‘quotient’ and ‘remainder’ used in dividing polynomials  use the method of equating coefficients  use the remainder theorem and the factor theorem.

1.2.1 Polynomials, adding, subtracting, multiplying and dividing Polynomials = sum of all the terms of axn, where n = 0, 1, 2,….. a = coefficients n = degree of the polynomial 0 = constant polynomial 1 = linear 2 = quadratic 3 = cubic 4 = quartic

Example 1: (Page 6, Q4d) f(x) = x3 – 2x2 + 5x – 3 and g(x) = x2 – x + 4. Simplify 3f(x) – 2g(x). Example 2: (p.6, Q5l) Evaluate (x2 + 1)(x – 3)(2x2 – x + 1).

1.2.2 Polynomial identities Example 3: (P. 6, Q7h) Find the values of A and B for the following polynomial. (Ax + B)(3x2 – 2x – 1) ≡ 6x3 – 7x2 + 1 Example 4: Given that for all real values of x, 2x3 + 3x2 – 14x – 5 ≡ (Ax + B)(x + 3)(x + 1) + C Find A, B, and C.

1.2.3 Division of a polynomial with a linear or a quadratic polynomial Example 5: Factorise x3 + 3x + 14 given that one factor is (x + 2). Example 6: Find the quotient and the remainder of x4 – x3 + 5x2 + 4x – 36 ÷ (x2 – x + 9). Example 7

1.3 Remainder theorem, Factor theorem, Solving polynomial equations 1.3.1 REMAINDER THEOREM When a polynomial f(x) is divided by x – a, the remainder R is f(a). Example 8: Find the remainder for 3x 4 + x2 – 7x + 6 ÷ (x + 3). 1.3.2 FACTOR THEOREM If x – a is a factor of f(x), then f(a) = 0 If f(a) = 0, then x – a is a factor of f(x). Example 9 Find one of the factors of the following polynomial by factor theorem: (a) 2x3 – 9x2 + 7x + 6 (b) 2x3 + 3x2 – 8x + 3 Example 10 Given that both (x – 1) and (x – 2) are factors of 2x3 – x2 + ax + b, find the values of a and b.

Ex 2: Dividing a polynomial using a quadratic expression, remainder and factor theorem 1. Factorise the following polynomial with a given factor: (a) x4 – 5x – 1 with factor (x2 – x – 2) (b) x4 + x3 – 12x2 – 4x + 16 with factor (x2 + 2x – 8) (c) 6x3 + 5x2 – 17x – 6 with factor (3x2 + 7x + 2) (d) 2x3 – x2 – 15x + 18 with factor (x2 + x – 6) 2. Using factor theorem solve the following questions: (a) Determine if any of the following expression is a factor of 6x4 – x3 – 21x2 – 6x + 8: (i) x – 1 (ii) x + 2 (iii) x – 2 (b) Find a factor for each of the following functions:(i) 4x4 - 4x3 – 9x2 + x + 2 (ii) x4 – x3 + 5x2 + 4x – 36 (iii) x3 – 2x2 – 7x + 12 3. Given that f(x) = x3 + ax + b. If (x – 1) is a factor of f(x) and the remainder is – 6 when f(x) is divided by (x + 1), find the values of constants a and b. 4. Given that 6x3 + x2 – 29x + d is divisible by both 2x – 1 and x + 3, find the values of c and d. 5. The cubic polynomial 2x3 + 5x2 + ax – 6, where a is a constant, is denoted by f(x). Given that (x + 2) is a factor of f(x), find the value of a.

1.3.3 Solving polynomial equations Example 1

Solve the equation 2x3 – 7x2 + 9 = 0

Example 2

Solve the equation 2x3 = 9x2 – 11x + 2, giving your answer correct to 2 decimal places where necessary.

Exercise 3: Solving polynomial equations 1. Solve the following equations: (a) 2x3 = x2 + 5x + 2 (b) x3 + 16 = 12x (c) 2x3 + 5x2 = 2 – x (d) 4x3 + 3x2 – 16x = 12 (e) 2x3 – 11x = 6 – 3x2 (f) x3 + 4 = x (x + 4) 4 2 (g) x  3  x (h) x(x + 3)(x – 1) = x + 8 2. Solve the following equations, giving your answer correct to 2 decimal places where necessary. (a) x3 + x2 – 8x + 4 = 0 (b) 3x3 = 5x + 2 (c) 8x + 6 = x3 + x2 (d) 2x2 + 7x = x3 + 12 (e) 3x3 + 5x2 = 3x + 2 (f) 2x3 + 6x – 6 = (13x – 6)(x – 1) 3. Given that P(x) = x4 + ax3 – x2 + bx – 12 has factors x – 2 and x + 1, solve the equation P(x) = 0. 4. Given f(x) = 2x3 + ax2 – 7a2x – 6a3, determine whether or not x – a and x + a are factors of f(x). Hence find, in terms of a, the roots of f(x) = 0.