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Chapter 5: Problem 1

Divide as indicated. Check each answer by showing that the product of the divisor and the quotient, plus the remainder, is the dividend. $$\frac{x^{2}+6 x+8}{x+2}$$

Short Answer

Expert verified
The quotient of the division is \(x+4\). Verification by multiplication of the quotient and the divisor equals the original dividend proved our solution to be correct.

Step by step solution

01

Factor the Quadratic Polynomial

Firstly, factorise the quadratic polynomial in the numerator. The polynomial \(x^{2}+6x+8\) factors into \( (x+2)(x+4) \). So, the division expression becomes \( \frac{(x+2)(x+4)}{x+2} \).
02

Perform the Polynomial Division

Now, cancel out the common factors in the numerator and the denominator which is \(x+2\) in this case. This leaves us with \(x+4\). So, the quotient of the division is \(x+4\).
03

Verify the Result

Lastly, check the result. The product of the divisor \(x+2\) and the quotient \(x+4\) should be the same as the original dividend \(x^{2}+6x+8\). In this case, it does (\((x+2)(x+4) = x^{2}+6x+8\)) , which verifies our solution.

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Most popular questions from this chapter

Use a vertical format to find each product. $$\begin{array}{l}x^{2}+7 x-3 \\\x^{2}-x-1 \\\\\hline\end{array}$$

In each exercise, find the product. $$9 x y\left(3 x y^{2}-y+9\right)$$

Are the expressions $$ \frac{12 x^{2}+6 x}{3 x} \text { and } 4 x+2 $$ equal for every value of \(x ?\) Explain.

Perform the indicated operations. $$(x+4)(x-5)-(x+3)(x-6)$$

In Exercises \(53-78,\) divide the polynomial by the monomial. Check each answer by showing that the product of the divisor and the quotient is the dividend. $$\frac{8 x^{6} y^{3}-12 x^{8} y^{2}-4 x^{14} y^{6}}{-4 x^{6} y^{2}}$$
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