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Chapter 2: Problem 83

Simplify the rational expression by using long division or synthetic division. $$\frac{x^{4}+6 x^{3}+11 x^{2}+6 x}{x^{2}+3 x+2}$$

Short Answer

Expert verified
The simplified expression of \(\frac{x^{4}+6 x^{3}+11 x^{2}+6 x}{x^{2}+3 x+2}\) is \(x^{2}+3x+2\).

Step by step solution

01

Identify The Divisor and Dividend

In this exercise, the polynomial in the numerator \(x^{4}+6 x^{3}+11 x^{2}+6 x\) is your dividend while the polynomial in the denominator \(x^{2}+3 x+2\) is your divisor.
02

Perform Polynomial Division Using Long Division

The procedure is similar to arithmetic long division. Start by dividing the first term in the dividend \(x^{4}\) by the first term in the divisor \(x^{2}\) to get \(x^{2}\), which is your first term of the quotient. Multiply the entire divisor by the new term found and subtract that from the dividend to find the remainder. The new dividend to continue the process is now the calculated remainder. Repeat this process until the degree of the remainder is less than the degree of the original divisor.
03

Write Down the Final Simplified Expression

Resulting from the polynomial long division, the quotient obtained: \(x^{2}+3x+2\) is the simplified expression.

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Most popular questions from this chapter

Decide whether the statement is true or false. Justify your answer. It is possible for a third-degree polynomial function with integer coefficients to have no real zeros.

A rectangular playing field with a perimeter of 100 meters is to have an area of at least 500 square meters. Within what bounds must the length of the rectangle lie?

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{1}{4} x^{2}-x+\frac{1}{4}=\frac{1}{4}\left(4 x^{3}-x^{2}-4 x+1\right)$$

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-4 x^{3}+5 x^{2}-2 x-6\) (Hint: One factor is \(\left.x^{2}-2 x-2 .\right)\)

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$x^{2}+b x-4=0$$
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