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Chapter 2: Problem 14

Use long division to divide. $$\left(6 x^{3}-16 x^{2}+17 x-6\right) \div(3 x-2)$$

Short Answer

Expert verified
The solution to the polynomial division is \(2x^2 - 6x + 3\).

Step by step solution

01

Divide the first terms

To begin with, divide the leading term of the numerator, 6x^3, by the leading term of the denominator, 3x. This will give a quotient of 2x^2.
02

Multiply the entire denominator by the quotient and subtract the result from the original polynomial

Next, multiply the whole divisor (3x - 2) by the quotient obtained in step 1, 2x^2. Then subtract the result from the original polynomial.
03

Drop down the next term

Bring down the next term from the original polynomial (which is +17x after the subtraction in step 2)
04

Repeat from Step 1

Following the repetition, another quotient term is obtained as -6x. Repeat step 2 and 3 then.
05

Repeat till all terms are exhausted

Keep repeating the cycle until all terms from the original polynomial are included. After the cycle is finished, one last quotient term of +3 is obtained.
06

Write out the result

The result of the long division is the sum of the quotient terms obtained - it represents the result of the polynomial division.

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Most popular questions from this chapter

Explore transformations of the form \(g(x)=a(x-h)^{5}+k\) (a) Use a graphing utility to graph the functions \(y_{1}=-\frac{1}{3}(x-2)^{5}+1\) and \(y_{2}=\frac{3}{5}(x+2)^{5}-3\) Determine whether the graphs are increasing or decreasing. Explain. (b) Will the graph of \(g\) always be increasing or decreasing? If so, then is this behavior determined by \(a, h,\) or \(k ?\) Explain. (c) Use the graphing utility to graph the function \(H(x)=x^{5}-3 x^{3}+2 x+1\) Use the graph and the result of part (b) to determine whether \(H\) can be written in the form \(H(x)=a(x-h)^{5}+k\) Explain.

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Simplify the complex number and write it in standard form. $$(-i)^{6}$$

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=x^{3}-4 x^{2}+1\) (a) Upper: \(x=4\) (b) Lower: \(x=-1\)
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