Question

70 pointsss for the correct answer

When the function f(x) is divided by 4x−5, the quotient is 3x^2 −x+3 and the remainder is −2. Find the function f(x) and write the result in standard form.

Answer 1

f(x) = 12x³ - 19x² + 17x - 17

Explanation:

We can the numerator by multiplying the denominator by the quotient, and then adding the remainder.

We're given a denominator of 4x - 5

a quotient of 3x² - x + 3

and a remainder of -2

so the original numerator is:

(4x - 5)(3x² - x + 3) - 2

= 12x³ - 4x² + 12x - 15x² + 5x - 15 - 2

= 12x³ - 19x² + 17x - 17

Let's test that by dividing it by 4x -5 with long division:

`\ \ \ \ \ \ \ \ \ \ 3x^2 - x + 3\\4x - 5\overline{)12x^3 - 19x^2 + 17x - 17}\\.\ \ \ \ \ \ \ \ \ \underline{2x^3 - 15x^2}\\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ -4x^2 + 17x\\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{-4x^2 + 5x} \\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 12x - 17\\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{12x - 15}\\.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2R`

That matches, so we know our answer's right, and:

f(x) = 12x³ - 19x² + 17x - 17