Verify division algorithm for the polynomials $f (x) = 8 + 20 x + x^{2} - 6 x^{3}$ and $g (x) = 2 + 5 x - 3 x^{2}$

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Solution

We can write $f (x) as - 6 x^{3} + x^{2} + 20 x + 8$ and $g (x) as - 3 x^{2} + 5 x + 2$

Quotient = $2 x + 3$
Remainder = $x + 2$

By using division rule, we have

Divided = Quotient × Divisor + Remainder

$∴ - 6 x^{3} + x^{2} + 20 x + 8 = (- 3 x^{2} + 5 x + 2) (2 x + 3) + x + 2 \Rightarrow - 6 x^{3} + x^{2} + 20 x + 8 = - 6 x^{3} + 10 x^{2} + 4 x - 9 x^{2} + 15 x + 6 + x + 2 \Rightarrow - 6 x^{3} + x^{2} + 20 x + 8 = - 6 x^{3} + x^{2} + 20 x + 8$

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