Question

Divide the polynomial p(x) = 3x4 + 4x3 +4x2 -8x+1 by q(x) = 3x+1. Also find what should be

added to ‘x’ so that it is completely divisible by q (x) .

Answer 1

$p(x) = 3x {}^{4} + 4x {}^{3} + 4x {}^{2} - 8x + 1 \\ q(x) = 3x + 1 \\ by \: remainder \: theorem \\ 3x + 1 = 0 \\ 3x = - 1 \\ x = \frac{ - 1}{3} \\ so \: remainder = p( \frac{ - 1}{3} ) \\ = 3 \times ( \frac{ - 1}{3}) {}^{4} + 4 \times ( \frac{ - 1}{3} ) {}^{3} \\ + 4 \times ( \frac{ - 1}{3} ) {}^{2} - 8 \times \frac{ - 1}{3} + 1 \\ = 3 \times \frac{1}{81} - 4 \times \frac{1}{27} + 4 \times \frac{1}{9} \\ - 8 \times \frac{ - 1}{3} + 1 \\ = \frac{1}{27} - \frac{4}{27} + \frac{4}{9} + \frac{8}{3} + 1 \\ = \frac{1 - 4 + 12 + 72 + 27}{27} \\ = \frac{108}{27} = 4$
Remainder 4 is left.
So,if we add (-4) to p(x) then it will completely divisible by q(x).

Answer 2

Answer:

Here's your answer.. ⬇⬇

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quotient (x) = x³ + x² + x - 3

remainder (x) = 4

Adding ( - 4 ) to p(x) we get..,

p(x) = 3x^4 + 4x^3 + 4x^2 - 8x - 3 which is divisible by q(x).

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Hope it helps..

Thanks :)

Step-by-step explanation: