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Chapter 5: Problem 57

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(4 x^{2}+25 x-3\) is divided by \(4 x+1,\) the remainder is 9.

Short Answer

Expert verified
Upon evaluating the division, the remainder is not 9 but rather -23. Thus, the statement given in the problem is false. The corrected version would be: 'If \(4 x^{2}+25 x-3\) is divided by \(4 x+1\), the remainder is -23.'

Step by step solution

01

Evaluate the Division

Use polynomial long division or synthetic division to divide the polynomial \(4 x^{2}+25 x-3\) by \(4 x+1\).
02

Check the Remainder

Upon performing the division, observe the remainder that you get. If the remainder is 9, then the original statement is correct.
03

Correct the Statement if False

If the remainder is not 9, this means the given statement is false. The corrected statement would indicate the correct remainder upon this division.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. In each exercise, find the indicated products. Then, if possible, state a fast method for finding these products. (You may already be familiar with some of these methods from a high school algebra course.) a. \((x+3)(x+4)\) b. \((x+5)(x+20)\)

Use a vertical format to find each product. $$\begin{array}{l}x^{2}+7 x-3 \\\x^{2}-x-1 \\\\\hline\end{array}$$

In Exercises \(105-106,\) find the missing coefficients and exponents designated by question marks. $$\frac{3 x^{14}-6 x^{12}-7 x^{7}}{2 x^{7}}=-x^{7}+2 x^{5}+3$$

In Exercises \(53-78,\) divide the polynomial by the monomial. Check each answer by showing that the product of the divisor and the quotient is the dividend. $$\frac{20 x^{7} y^{4}-15 x^{3} y^{2}-10 x^{2} y}{-5 x^{2} y}$$

Exercises \(110-112\) will help you prepare for the material covered in the next section. In each exercise, perform the long division without using a calculator, and then state the quotient and the remainder. $$2 4 \longdiv { 2 9 5 8 }$$
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