Question

Divide 30x4 + 11x3 - 82x2 + 12x + 48 by (3x2 + 2x - 4) and verify the result by division algorithm.

Answer 1

$\textbf{Given:}$

$30x^4+11x^3-82x^2+12x+48{\div}3x^2+2x-4$

$\textbf{To find:}$

$\text{Quotient and remainder}$

$\textbf{Solution:}$

$\text{Consider,}$

$\begin{array}{r|l}&10x^2-3x-12\\\cline{2-2}3x^2+2x-4&30x^4+11x^3-82x^2+12x+48\\&30x^4+20x^3-40x^2\\\cline{2-2}&****-9x^3-42x^2+12x\\&****-9x^3-\;6x^2+12x\\\cline{2-2}&********-36x^2+\;0x+48\\&********-36x^2-12x+48\\\cline{2-2}\\&**************12x\\\cline{2-2}\end{array}$

$\text{From the above long division,}$

$\text{Quotient}=10x^2-3x-12$

$\text{Remainder}=12x$

$\textbf{Verification:}$

$\boxed{\textbf{Dividend=(Divisor \times Quotient)+Remainder}}$

$\textbf{(Divisor \times Quotient)+Remainder}$

$=(3x^2+2x-4)(10x^2-3x-12)+12x$

$=(30x^4-9x^3-36x^2+20x^3-6x^2-24x-40x^2+12x+48)+12x$

$=(30x^4+11x^3-82x^2+0x+48)+12x$

$=30x^4+11x^3-82x^2+12x+48$

$=\textbf{Dividend}$

$\textbf{Hence verified}$

Answer 2

Answer:

p(x)=30x^4+11x^3-82^2-12x+48

g(x)=3x^2+2x-4

After long division,

q(x)=10x^2-3x-12

r(x)=0

By division algorithm,

p(x)=g(x)*q(x)+r(x)

=(3x^2+2x-4)(10x^2-3x-12)+0

=30x^4+11x^3-82x^2-12x+48

Hence Verified