Question
Verify division algorithm for the polynomials $f(x) = 8 + 20x + x^2 - 6x^3$ and $g(x) = 2 + 5x - 3x^2$
✓
Answer
We can write $f(x)$ as $-6x^3 + x^2 + 20x + 8$ and
$g(x)$ as $-3x^2 + 5x + 2$
Quotient $= 2x + 3$
Remainder$ = x + 2$ By using division rule,
we have Divided = Quotient $\times$ Divisor + Remainder
$\therefore$ $-6x^3 + x^2 + 20x + 8 = (-3x^2 + 5x + 2)(2x + 3) + x + 2 $
$\Rightarrow -6x^3 + x^2 + 20x + 8 = -6x^3 + 10x^2 + 4x - 9x^2 + 15x + 6 + x + 2$
$ \Rightarrow -6x^3 + x^2 + 20x + 8 = -6x^3 + x^2 + 20x + 8$
$g(x)$ as $-3x^2 + 5x + 2$
Quotient $= 2x + 3$
Remainder$ = x + 2$ By using division rule,
we have Divided = Quotient $\times$ Divisor + Remainder
$\therefore$ $-6x^3 + x^2 + 20x + 8 = (-3x^2 + 5x + 2)(2x + 3) + x + 2 $
$\Rightarrow -6x^3 + x^2 + 20x + 8 = -6x^3 + 10x^2 + 4x - 9x^2 + 15x + 6 + x + 2$
$ \Rightarrow -6x^3 + x^2 + 20x + 8 = -6x^3 + x^2 + 20x + 8$
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