Question

Divide 9x ^ 3 + 18x ^ 2 - 13x + 5 by 3x - 1 using division and write the division in the form P = DQ + R

Answer 1

Answer:

$\frac{9x^3\:+\:18x\:^2\:-\:13x\:+\:5}{3x-1}=9x^3+18x^2-13x+5$

Step-by-step explanation:

DIVISION ALGORITHM: If $p(x)$ and $d(x)\neq 0$  are polynomials, and the degree of $d(x)$ is less than or equal to the degree of $f(x)$,  then there exist unique polynomials $q(x)$ and $r(x)$, so that

                                               $\frac{p(x)}{d(x)} =q(x)+\frac{r(x)}{d(x)}$

and so that the degree of $r(x)$  is less than the degree of $d(x)$.

To find $\frac{9 x^{3} + 18 x^{2} - 13 x + 5}{3 x - 1}$ you must:

$\mathrm{Divide\:the\:leading\:coefficients\:of\:the\:numerator\:}9x^3+18x^2-13x+5\\\mathrm{and\:the\:divisor\:}3x-1\mathrm{\::\:}\frac{9x^3}{3x}=3x^2$

$\mathrm{Quotient}=3x^2$

$\mathrm{Multiply\:}3x-1\mathrm{\:by\:}3x^2:\:9x^3-3x^2\\\mathrm{Subtract\:}9x^3-3x^2\mathrm{\:from\:}9x^3+18x^2-13x+5\mathrm{\:to\:get\:new\:remainder}$

$\mathrm{Remainder}=21x^2-13x+5$

Therefore,

$\frac{9x^3+18x^2-13x+5}{3x-1}=3x^2+\frac{21x^2-13x+5}{3x-1}$

$\mathrm{Divide}\:\frac{21x^2-13x+5}{3x-1}$

$\mathrm{Divide\:the\:leading\:coefficients\:of\:the\:numerator\:}21x^2-13x+54\\\mathrm{and\:the\:divisor\:}3x-1\mathrm{\::\:}\frac{21x^2}{3x}=7x\\\\\mathrm{Quotient}=7x$

$\mathrm{Multiply\:}3x-1\mathrm{\:by\:}7x:\:21x^2-7x\\\mathrm{Subtract\:}21x^2-7x\mathrm{\:from\:}21x^2-13x+5\mathrm{\:to\:get\:new\:remainder}\\\\\mathrm{Remainder}=-6x+5$

Therefore,

$\frac{21x^2-13x+5}{3x-1}=7x+\frac{-6x+5}{3x-1}\\\\\frac{9x^3+18x^2-13x+5}{3x-1}=3x^2+7x+\frac{-6x+5}{3x-1}$

$\mathrm{Divide}\:\frac{-6x+5}{3x-1}$

$\mathrm{Divide\:the\:leading\:coefficients\:of\:the\:numerator\:}-6x+5\\\mathrm{and\:the\:divisor\:}3x-1\mathrm{\::\:}\frac{-6x}{3x}=-2\\\\\mathrm{Quotient}=-2$

$\mathrm{Multiply\:}3x-1\mathrm{\:by\:}-2:\:-6x+2\\\mathrm{Subtract\:}-6x+2\mathrm{\:from\:}-6x+5\mathrm{\:to\:get\:new\:remainder}\\\\\mathrm{Remainder}=3$

Therefore,

$\frac{-6x+5}{3x-1}=-2+\frac{3}{3x-1}\\\\\frac{9x^3+18x^2-13x+5}{3x-1}=3x^2+7x-2+\frac{3}{3x-1}$