Question

7. Find {f/g)(x) if f(x)=7x³-5 x²+42x-30 and g(x)=7x-5.

{f/g)(x)=x²+6
(f/g)(x)=7x²+6
(f/g)(x)=x²-6
(f/g)(x)=7x²-6

Answer 1

First choice:
`x^2 + 6`

Explanation:

`(f)/(g)(x) \;is\;nothing\;but\;(f(x))/(g(x))`

So you could divide
7x³-5 x²+42x-30 by 7x-5 and arrive at the correct answer.

But polynomial division is a pain

So instead we can multiply 7x -5 with each of the choices given and see which one will result in 7x³-5 x²+42x-30

But there is an easier way

• We can eliminate the second choice, 7x³+ 6 and last choice 7x³ 6. That is because there is a x³ term in the polynomial and its coefficient is 7

• Both of these incorrect choices will result in 7x² · 7x which is 49x³. So eliminate these two.

The first and third choices differ only in the sign of the constant
The second choice has +6 as the constant and the third has -6

Therefore the constant in the product should be either 6 · (-5) = -30 or

6(5) = 30

Since the coefficient of f(x) is -30, the correct answer is the first choice

`x^2 + 6`

Verify:

`\left(x^2+6\right)\left(7x-5\right)\\\\=x^2\cdot \:7x+x^2\left(-5\right)+6\cdot \:7x+6\left(-5\right)\\\\=x^2\cdot \:7x+x^2\left(-5\right)+6\cdot \:7x+6\left(-5\right)\\\\=7x^3-5x^2+42x-30`