Power Functions, Polynomials, Remainder And Factor Theorem

University:
Stanford University
Course:
Introduction to Mathematical Thinking
Academic year:

2023
Views:

116

Pages:

3
Author:

MAlphen

Power Functions 4 1. Sketch the graph pf 𝑦 = √𝑥 for 𝑥 > 0. (b) Find the coordinates of the point(s) of intersection of 𝑦 = Ans: 4 and 𝑦2 = 81𝑥. √𝑥 Polynomials, Remainder and Factor Theorem 1. Given that 𝑓(𝑥) = 𝑎𝑥3 + 𝑏𝑥2 + 2 leaves a remainder of −2 when divided by (𝑥 + 2) and 3𝑓(𝑥) + 6 is exactly divisible by (𝑥 − 1). Find the value of 𝑎 and 𝑏. b. (i) Factorise 6𝑥3 − 7𝑥2 − 8𝑥 + 5 completely (ii) Hence, solve the equation 5𝑥3 − 8𝑥2 − 7𝑥 + 6 = 0. Ans: (a) 𝑎 = −1, 𝑏 = −3 (b)(i) (𝑥 − 1)(2𝑥 − 1)(3𝑥 − 5) (ii) 𝑥 = −1. 𝑥 = 2, 𝑥 = 3 5 2. Given that 3𝑥3 − 8𝑥2 + 4𝑥 − 5 = 𝑥(𝐴𝑥 + 𝐵)(𝑥 − 2) + 𝐶 for all values of 𝑥, find the values of A,B and C. The cubic polynomial f(x) is such that the coefficient of 𝑥3 is 2 and the root of the equation 𝑓(𝑥) = 0 are −1, 3 and k. Given that 𝑓(𝑥) has a remainder of 20 when divided by 𝑥 − 4, find (i) the value of k (ii) the remainder when 𝑓(𝑥) is divided by 𝑥 + 2. Ans: (a) A=3, B= -2, C=-5 (b)(i) 𝑘 = 2 (ii) −40 3. The function 𝑓 is defined by 𝑓(𝑥) = 2𝑥3 + 4𝑎𝑥2 − 6𝑥2 + (10 − 15𝑎)𝑥 + 11𝑎 − 6 = 0, where 𝑎 is a constant. (i) Show that 𝑥 − 1 is a factor of 𝑓(𝑥). (ii) Find the possible values of 𝑎 for which the equation 𝑓(𝑥) = 0 only has exactly 2 real roots. Ans: (ii) 𝑎 = −4, 𝑎 = 0