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Chapter 5: Problem 71

Divide the polynomial by the monomial. Check each answer by showing that the product of the divisor and the quotient is the dividend. $$\frac{25 x^{7}-15 x^{5}-5 x^{4}}{5 x^{3}}$$

Short Answer

Expert verified
The quotient after the division is \(5x^4 - 3x^2 - x\). The product of the divisor and the quotient is indeed the original dividend, hence validating that the quotient is accurate.

Step by step solution

01

Divide each term

Start by dividing each term in the numerator by \(5x^3\). This results in the polynomial \(5x^4 - 3x^2 - x\).
02

Verify the result

Now, we will verify our answer by multiplying the divisor and the quotient: \(5x^3 * (5x^4 - 3x^2 - x)\). If we did everything correctly, this should result in the original dividend \(25 x^{7}-15 x^{5}-5 x^{4}\).
03

Multiply

Carry out the multiplication and simplify the result. This gives us the polynomial \(25 x^{7}-15 x^{5}-5 x^{4}\).

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Most popular questions from this chapter

Find the absolute value: \(|-20.3|\)

Subtract: \(-6.4-(-10.2) .\) (Section 1.6, Example 2)

In Exercises \(156-163\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\left(4 \times 10^{3}\right)+\left(3 \times 10^{2}\right)=4.3 \times 10^{3}$$

perform the indicated operations. $$\begin{aligned} &\left[\left(7 y^{2}-4 y+2\right)-\left(12 y^{2}+3 y-5\right)\right]-\left[\left(5 y^{2}-2 y-8\right)+\left(-7 y^{2}+10 y-13\right)\right] \end{aligned}$$

Use a graphing utility to determine whether the divisions have been performed correctly. Graph each side of the given equation in the same viewing rectangle. The graphs should coincide. If they do not, correct the expression on the right side by using polynomial division. Then use your graphing utility to show that the division has been performed correctly. $$\frac{x^{2}-4}{x-2}=x+2$$
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