Question

1. If x 2 + 2x -3 is a factor of the polynomial f(x) = 2x 3 + ax 2 + ax – b, where a and b are constants, find the values of a and b.

(b) Hence:
(i) Factorize f(x) completely
(ii) State the zeros of f(x)
(iii) Find the remainder when f(x) is divided by (x-5)
2. Find the equation whose roots are 1, 2 and 3.
3. Given the equation 4(y 2 – 2)2 = k(y – 1) , where k is constant.
(i)Show that the sum of roots equals their product
(ii) Find the value of k for which the difference between their roots is 3¾

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Answer 1

Hi,

Explanation:

1.
If x 2 + 2x -3 is a factor of the polynomial f(x) = 2x 3 + ax 2 + ax – b, where a and b are constants, find the values of a and b.

`x^2+2x-3=0\\x^2+2x+1-4=0\\(x+1)^2-4=0\\(x+1+2)(x+1-2)=0\\(x+3)(x-1)=0\\x=-3\ or\ x=1\\`

Let's divide f(x) by x-1 and then by x+3

`\begin{array}c&x^3&x^2&x&1\\----&---&---&---&---\\&2&a&a&-b\\x=1&&2&a+2&2a+2\\----&---&---&---&---\\&2&a+2&2a+2&2a+2-b=0\\x=-3&&-6&-3a+12&\\----&---&---&---&---\\&2&a-4&-a+14=0&\\\end {array}\\\\`

`\left\{\begin{array}{ccc}2a+2-b&=&0\\-a+14&=&0\\\end {array} \right.\\\\\\\left\{\begin{array}{ccc}a&=&14\\b&=&30\\\end {array} \right.\\`

(i) Factorize f(x) completely

`2x^3+14x^2-14x-30\\=(x-1)(x+3)(2x+10)\\\\\boxed{f(x)=2(x-1)(x+3)(x+5)}\\\\`

(ii) State the zeros of f(x)

Zeros are 1,-3,-5

(iii) Find the remainder when f(x) is divided by (x-5)

remainder is 0

2.

Find the equation whose roots are 1, 2 and 3.

E(x)=(x-1)(x-2)(x-3)=

3. Given the equation 4(y² - 2)² = k(y – 1) , where k is constant.