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Synthetically Divide: (2x^3+x^2-22x+20)/(2x-3)

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Synthetically Divide: (2x^3+x^2-22x+20)/(2x-3)

Divide by synthetic division:

$(2x^3+x^2-22x+20)\div (2x-3)$

Begin by solving for

$x$ in the divisor:

$2x-3=0$

$\displaystyle x=\frac{3}{2}$

Now perform synthetic division:

Given a remainder of

$4$ , write the expression as:

$\displaystyle 2x^2+4x-16-\frac{4}{x-\frac{3}{2}}$

Recall that you set the divisor

$2x-3$ equal to

$0$ at the start. Since

$x$ has a coefficient of

$2$ , divide the expression above by

$2$ as well.

$\displaystyle \frac{\left(2x^2+4x-16-\frac{4}{x-\frac{3}{2}}\right)}{2}$

Simplify by finding a common denominator in the numerator:

$\displaystyle =\frac{\frac{\left(2x^2+4x-16\right)\left(x-\frac{3}{2}\right)-4}{x-\frac{3}{2}}}{2}$

$\displaystyle =\frac{\left(2x^2+4x-16\right)\left(x-\frac{3}{2}\right)-4}{x-\frac{3}{2}}\div 2$

$\displaystyle =\frac{\left(2x^2+4x-16\right)\left(x-\frac{3}{2}\right)-4}{x-\frac{3}{2}}\times \frac{1}{2}$

Expand and multiply:

$\displaystyle = \frac{2x^3+x^2-22x+22}{2x-3}$

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