Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Appendix A - Review - A.3 Polynomials - A.3 Assess Your Understanding - Page A30: 69

Answer

Quotient $=x^2-2x+\dfrac{1}{2}$ Remainder $=\dfrac{5x+1}{2}$

Work Step by Step

The given expression is $(2x^4-3x^3+x+1)\div(2x^2+x+1)$ Rewrite the expression as $(2x^4-3x^3+0x^2+x+1)\div(2x^2+x+1)$ Perform long division: $\begin{matrix} & x^2 & -2x & +1/2 ​& & && \leftarrow &\text{Quotient}\\ &-- &-- &--&--& \\ 2x^2+x+1) &2x^4&-3x^3&+0x^2&+x&+1 & \\ ​& 2x^4 &+x^3 & +x^2& &&& \leftarrow &x^2(2x^2+x+1) \\ & -- & -- & --& &&&\leftarrow &\text{subtract} \\ & 0 & -4x^3& -x^2 &+x & & \\ ​& &-4x^3 & -2x^2&-2x &&& \leftarrow &-2x(2x^2+x+1) \\ & & -- & --& -- &&&\leftarrow &\text{subtract} \\ & & 0 & +x^2& +3x &+1 & & \\ ​& &&x^2 & +x/2&+1/2 && \leftarrow &1/2(2x^2+x+1) \\ & && -- & --& -- &&\leftarrow &\text{subtract} \\ & && 0 & +5x/2& +1/2 & & \leftarrow & \text{Remainder} ​\end{matrix}$ Checking: $\text{(Quotient)(divisor)+ Remainder}$ $=(x^2-2x+1/2)(2x^2+x+1)+5x/2+1/2$ $=2x^4-4x^3+x^2+x^3-2x^2+x/2+x^2-2x+1/2+5x/2+1/2$ $=2x^4-3x^3+x+1$ $=\text{ Dvidend}$ Hence, the quotient is $x^2-2x+\frac{1}{2}$ and the remainder is $\dfrac{5x+1}{2}$.
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