Verify the division algorithm for the polynomials $p (x) = 2 x^{4} - 6 x^{3} + 2 x^{2} - x + 2 and g (x) = x + 2$ .

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Solution

$p (x) = 2 x^{4} - 6 x^{3} + 2 x^{2} - x + 2$ and $g (x) = x + 2$

Quotient = $2 x^{3} - 10 x^{2} + 22 x - 45$

Remainder = 92

Verification:

Divisor × Quotient + Remainder
$= (x + 2) \times (2 x^{3} - 10 x^{2} + 22 x - 45) + 92 = x (2 x^{3} - 10 x^{2} + 22 x - 45) + 2 (2 x^{3} - 10 x^{2} + 22 x - 45) + 92 = 2 x^{4} - 10 x^{3} + 22 x^{2} - 45 x + 4 x^{3} - 20 x^{2} + 44 x - 90 + 92 = 2 x^{4} - 6 x^{3} + 2 x^{2} - x + 2$
= Dividend

Hence verified.

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p (x) = 2 x^{2} - 9 x + 9; g (x) = x - 3

In each of the following, using the remainder Theorem, find the remainder when f(x) is divided by g(x) and verify the result y actual division.

f(x) = $2 x^{4} - 6 x^{3} + 2 x^{2} - x + 2$ , g(x) = x + 2

P(x) =2x⁴-6x³+2x²-x+2

G(x)=x+2

Find remainder by long division.

f (x) = 2 x^{4} - 6 x^{3} + 2 x^{2} - x + 2

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