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Chapter 3: Problem 14

Use long division to divide. $$\left(4 x^{3}-7 x^{2}-11 x+5\right) \div(4 x+5)$$

Short Answer

Expert verified
The solution to the problem is \(x^{2}-\frac{3}{4}x-\frac{1}{16} + \frac{ \frac{21}{16}}{4x+5}\).

Step by step solution

01

Setup the Long Division

To set up the long division, write the dividend \(4x^{3}-7x^{2}-11x+5\) inside the division symbol and the divisor \(4x+5\) outside the division symbol, similar to traditional long division.
02

Divide the First Term

Divide the first term of the dividend \(4x^3\) by the first term of the divisor \(4x\). This gives us \(x^2\), which is the first term of the quotient.
03

Multiply and Subtract

Multiply the divisor \(4x+5\) by the first term of the quotient \(x^2\), then subtract this from the original dividend. This gives a new expression of \(-3x^{2}-11x+5\).
04

Repeat the Process

Repeat step 2 with the first term of the new expression and the first term of the divisor. So, \(-3x^2\) divided by \(4x\) gives \(-\frac{3}{4}x\) which is added to the quotient. Substract to get a new expression \( -\frac{1}{4}x +5\).
05

Repeat the Process Again

Repeat step 2 again with the first term of the new expression and the first term of the divisor. So, \(-\frac{1}{4}x\) divided by \(4x\) gives \(-\frac{1}{16}\). After subtraction, we get \( \frac{21}{16}\). Since the degree of this term is less than the degree of the divisor, the division process stops here.
06

Write the Final Result

Combine all terms of the quotient and the remainder to express the final result of the division. The quotient \(x^{2}-\frac{3}{4}x-\frac{1}{16}\) and the remainder \( \frac{21}{16}\) , and the divisor \(4x+5\) makes the final result as \(x^{2}-\frac{3}{4}x-\frac{1}{16} + \frac{ \frac{21}{16}}{4x+5}\).

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Most popular questions from this chapter

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f .\) Then find all real zeros of the function. \(f(x)=x^{4}-4 x^{3}+16 x-16\) Upper bound: \(x=5\) Lower bound: \(x=-3\)

Let \(f(x)=14 x-3\) and \(g(x)=8 x^{2} .\) Find the indicated value. \((g-f)(3)\)

Find all real zeros of the polynomial function. $$f(z)=z^{4}-z^{3}-2 z-4$$

Write the general form of the equation of the line that passes through the points. $$(-6,1),(4,-5)$$

Solve the inequality and sketch the solution on the real number line. Use a graphing utility to verify your solution graphically. \(|x+8|-1 \geq 15\)
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