Question

if P(x) = x 3 + x ² – 2x+3 & g(x) = x+3, Find the
quotient and remainder on dividing P(x) by (x).
Also verify the divison algorithm​
Pls answer step by step

Answer 1

$\sf\large\underline{Given:}$

$\tt{\implies p(x)=x^3+x^2-2x+3}$

$\tt{\implies g(x)=x+3}$

$\sf\large\underline{To\: Find:}$

$\tt{\implies Remainder\: Quotient\:and\: Verify}$

$\sf\large\underline{Solution:}$

• Simply by dividing p(x) by g(x) to calculate remainder and quotient than verify it. With the help of formula]

$\begin{array}{c|ccccc}&x^2 &-2x&+4\\\cline{2-6}x+3&x^3&+x^2&-2x&+3\\ & x^3&+3x^2 \\ &-&-\\\cline{2-6} & & -2x^2&-2x&+3 \\ &&-2x^2&-6x\\ &&+&+\\\cline{2-6} &&&4x&+3& \\&&&4x&+12\\ &&&-&-\\\cline{2-6}&&&&-9\\\end{array}$

$\sf\large\underline{By\: solving\:we\: get\:here:}$

$\tt{\implies Remainder=-9}$

$\tt{\implies Quotient=x^2-2x+4}$

• To verify by applying formula here]

$\tt{\implies Dividend=Divisor*quotient+remainder}$

$\tt{\implies P(x)=g(x)*q(x)+r(x)}$

$\tt{\implies P(x)=(x+3)(x^2-2x+4)+(-9)}$

$\tt{\implies P(x)=x^3-2x^2+4x+3x^2-6x+12-9}$

$\tt{\implies P(x)=x^3+x^2-2x+3}$

$\sf\large\underline{Hence,\:Verified:}$