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Chapter 3: Problem 1

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$ \left(x^{2}+8 x+15\right) \div(x+5) $$

Short Answer

Expert verified
The quotient, \(q(x)\), is \(x + 3\), and the remainder, \(r(x)\), is 0.

Step by step solution

01

Arrange the polynomial in the correct order

First, write the given polynomials in the decreasing order of powers. For the polynomial in this problem it is already given in the correct order.
02

Divide

The polynomial division is similar to the standard long division with numbers, where \((x^{2}+8 x+15)\) is considered as the dividend and \((x+5)\) as the divisor. So, divide the first term of the dividend (\(x^{2}\)) by the first term of the divisor (x). The result, x, is the first term of our quotient.
03

Multiply and subtract

Multiply the divisor (\(x + 5\)) by the first term of the quotient (x) and subtract the result from the dividend. This gives a new polynomial, \(3x + 15\). This is the first subtraction step.
04

Continue the process

Continue the process of dividing the first term of the new polynomial by the first term of the divisor, and subtracting the result times the divisor from the new polynomial. This gives us a new term, 3, in the quotient.
05

Repeat the process until the degree of the remainder is less than the divisor

We repeat the process, and find that no more terms can be subtracted and the remainder is 0.

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Most popular questions from this chapter

Use long division to rewrite the equation for \(g\) in the form $$ \text {quotient}+\frac{\text {remainder}}{\text {divisor}} $$ Then use this form of the function's equation and transformations \( \text { of } f(x)=\frac{1}{x} \text { to graph } g \). $$ g(x)=\frac{2 x-9}{x-4} $$

In Exercises 94鈥�97, use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-2 x^{3}+6 x^{2}+3 x-1$$

Will help you prepare for the material covered in the next section. $$ \text { Solve: } 2 x^{2}+x=15 $$

Rewrite \(4-5 x-x^{2}+6 x^{3}\) in descending powers of \(x\).

There is more than one third-degree polynomial function with the same three x-intercepts.
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