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Consider the expressions $$\frac{4x^3+2x^2+6x+7}{2x+1}$$ and $$2x^2+3+\frac4{2x+1}.$$

a) Show that these expressions are equal when x=10

b) Explain why these expressions are not equal when x=-1/2

c) Show that these expressions are equal for all x  other than -1/2

In parts (a) and (c), begin by explaining what your strategy for solving will be.

Your first instinct on Part (c) may be to manipulate an equation until both sides are equal. However, this can confuse your reader: you would be writing equations that you don't know are true! Try to write your solution so that every equation you write is true.

May 15, 2019

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a) I will be substituting for x and simplifying.

$$\frac{4x^3+2x^2+6x+7}{2x+1} = 2x^2 + 3 + \frac{4}{2x+1}$$

$$\frac{4(1000)+2(100)+6(10)+7}{2(10)+1} = 2(100) + 3 + \frac{4}{2(10)+1}$$

$$\frac{4267}{21} = \frac{4267}{21}$$

(b) When plugging in $$x=-\frac{1}{2}$$ to the denominators $$2x+1$$ on both sides, we will end up dividing by zero, an undefined value.

(c) There's really two ways to do this. One is multiplying both sides by $$2x+1$$ and showing the results. The other is dividing $$4x^3+2x^2+6x+7$$ by $$2x+1$$ through long division to show that it equals $$2x^2+3+\frac{3}{2x+1}$$. However, I happen to hate long division with polynomials. So Ima do it tha otha way.

$$\frac{4x^3+2x^2+6x+7}{2x+1} = 2x^2 + 3 + \frac{4}{2x+1}$$

$$4x^3+2x^2+6x+7 = (2x+1) (2x^2 + 3 + \frac{4}{2x+1})$$

$$4x^3+2x^2+6x+7 = 4x^3+2x^2+6x+7$$

The right side multiplication looks intimidating, but it actually multiplies out that easily in one step.

Oh, and, see (b) for why it doesn't work for $$x = -\frac{1}{2}$$

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May 15, 2019