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Chapter 3: Problem 19

Divide using synthetic division. $$\left(3 x^{2}+7 x-20\right) \div(x+5)$$

Short Answer

Expert verified
The result of the division \(\left(3 x^{2}+7 x-20\right) \div(x+5)\) is \(3x - 8 + \frac{20}{x + 5}\).

Step by step solution

01

Setup Synthetic Division Table

Before starting, the problem should be set up in a synthetic division table format. Write down the coefficients of the dividend \(3x^{2}+7x-20\) which are 3, 7, and -20, respectively. The divisor is \(x+5\), so write -5 (as it's opposite of the constant in the divisor) to the left of the vertical bar in your synthetic division set up.
02

Perform Synthetic Division Process

First, 'drop' the first coefficient (3) down - this is essentially the freebie first step of synthetic division. Next, multiply this 'dropped down' number (3) by -5 (which is found from the divisor, \(x+5\)) to get -15. Write this under the next coefficient (7) and add them together to get -8. Repeat this process by multiplying -8 with -5 to get 40. Write this under the final coefficient (-20) and add them together to get 20.
03

Write Down the Final Answer

The number at the bottom of the synthetic division table is the remainder, and the other numbers are the coefficients of the quotient. The 3 (which was dropped down) will be the coefficient of \(x\). As \(x^{2}\) was divided by \(x\), the power decreases by one, becoming \(x\) in the coefficient. The value -8 will be the constant term of the solution. As a result, the division \( \left(3 x^{2}+7 x-20\right) \div(x+5)\) is \(3x - 8\) with a remainder of 20. So, the final answer is \(3x - 8 + \frac{20}{x + 5}\).

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Most popular questions from this chapter

What does it mean if two quantities vary inversely?

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degreeof \(p\) and \(q\) are as small as possible. More than one correct finction may be possible. Graph your function using a graphing utility to verify that it has the required propertics \(f\) has vertical asymptotes given by \(x=-2\) and \(x=2\) a horizontal asymptote \(y=2, y\) -intercept at \(\frac{9}{2}, x\) -intercepts at \(-3\) and \(3,\) and \(y\) -axis symmetry.

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degreeof \(p\) and \(q\) are as small as possible. More than one correct finction may be possible. Graph your function using a graphing utility to verify that it has the required propertics \(f\) has a vertical asymptote given by \(x=3,\) a horizontal asymptote \(y=0, y\) -intercept at \(-1,\) and no \(x\) -intercept.

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies directly as \(x . y=45\) when \(x=5 .\) Find \(y\) when \(x=13 .\)

Use the four-step procedure for solving variation problems given on page 424 to solve Exercises 1-10. \(y\) varies directly as \(x . y=65\) when \(x=5 .\) Find \(y\) when \(x=12 .\)
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