Emerson Barnes

2022-01-31

6.When a polynomial $x4+x3-10x2-1$ is divided by another polynomial, the quotient is $x3-3x2+2x-8$ and the remainder is 31. Find the other polynomial.

stamptsk

Expert

Given
Polynomial (dividend) ${x}^{4}+{x}^{3}-10{x}^{2}-1$
Remainder is 31
Polynomial (quotient) is ${x}^{3}-3{x}^{2}+2x-8$
To find the other polynomial(divisor)
Using division Formula
$\text{Dividend}=\text{quotient}×\text{divisor}+\text{remainder}$
Rewriting the formula
$⇒$ $\text{Dividend}-\text{remainder}=\text{quotient}×\text{divisor}$
$⇒$ $\frac{\text{Dividend}-\text{remainder}}{\text{quotient}}=\text{divisor}$
Hence
$\text{divisor}=\frac{\text{Divident}-\text{remainder}}{\text{quatient}}$
Plugin the known values
Divisor $=\frac{{x}^{4}+{x}^{3}-10{x}^{2}-1-31}{{x}^{3}-3{x}^{2}+2x-8}$
$⇒$ divisor $=\frac{{x}^{4}+{x}^{3}-10{x}^{2}-1-32}{{x}^{3}-3{x}^{2}+2x-8}$
Factorising numerator
$⇒$ divisor $=\frac{{x}^{3}-3{x}^{2}+2x-8\left(x+4\right)}{{x}^{3}-3{x}^{2}+2x-8}$
Canceling ${x}^{3}-3{x}^{2}+2x-8$ from numerator and denominator
$⇒$ divisor $=\left(x+4\right)$
Hence other polynomial (divisor) $=x+4$
Conclusion
Hence the other polynomial which gives quotient ${x}^{3}-3{x}^{2}+2x-8$ and remainder 31 on dividing polynomial${x}^{4}-{x}^{3}-10{x}^{2}-1$ is $x+4.$

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