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Chapter 5: Problem 35

Divide as indicated. Check each answer by showing that the product of the divisor and the quotient, plus the remainder, is the dividend. $$\frac{y^{4}-2 y^{2}+5}{y-1}$$

Short Answer

Expert verified
The quotient is \(y^{3}+y^{2}-y+4\).

Step by step solution

01

Polynomial Division

We perform polynomial division as we would do long division with numbers. Divide the first term in the numerator with the first term in the denominator, which is \(y^{4}\) by \(y\), and we obtain \(y^{3}\). Write \(y^{3}\) on your paper. Multiply the entirety of the denominator, \(y-1\), by \(y^{3}\) and subtract this from the original numerator to obtain a new numerator, which is \(y^{3}-2y^{2}+5\).
02

Repeat the Division Process

We repeat the process as in the first step. Divide \(y^{3}\) by \(y\), which is \(y^{2}\). Write this below \(y^{3}\). Multiply \(y-1\) by \(y^{2}\) and subtract this from the new numerator. The remainder is now \(-y^{2}+5\).
03

Further Repeat the Division Process

We further repeat the process as in the previous steps. Divide \(-y^{2}\) by \(y\), which is \(-y\). Write this below \(y^{2}\). Multiply \(y-1\) by \(-y\) and subtract this from the current numerator. This leaves us with the remainder of \(4\).
04

Final Division Process

We finally divide the last term \(4\) by \(y\). We cannot divide since the degree of denominator \(y-1\) is more than the degree of numerator \(4\). So, \(4\) is our remainder. We add this to our quotient along with the denominator to get the final polynomial division. Therefore, the quotient is \(y^{3}+y^{2}-y+4\).
05

Verify the Result

We now check the division by multiplying the divisor with the quotient and adding the remainder. So, the expression becomes \((y-1)(y^{3}+y^{2}-y)+4\). If it simplifies to the original numerator \(y^{4}-2y^{2}+5\), then the division is verified.

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Most popular questions from this chapter

In each exercise, find the product. $$9 x y\left(3 x y^{2}-y+9\right)$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. It is not possible to write a binomial with degree \(0 .\)

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Find each of the products in parts (a)-(c). a. \((x-1)(x+1)\) b. \((x-1)\left(x^{2}+x+1\right)\) c. \((x-1)\left(x^{3}+x^{2}+x+1\right)\) d. Using the pattern found in parts (a)-(c), find $(x-1)\left(x^{4}+x^{3}+x^{2}+x+1\right) without actually multiplying.

What polynomial, when divided by \(3 x^{2}\), yields the trinomial \(6 x^{6}-9 x^{4}+12 x^{2}\) as a quotient?
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