How do you divide $\frac{3 a^{3} + 17 a^{2} + 12 a - 5}{a + 5}$ $(3a^3+17a^2+12a-5)/(a+5)$ ?

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How do you divide $\frac{3 a^{3} + 17 a^{2} + 12 a - 5}{a + 5}$ $(3a^3+17a^2+12a-5)/(a+5)$ ?

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Answer 1 · 2017-07-14T04:37:58

$3 a^{2} + 2 a + 2 - \frac{15}{a + 5}$ $3a^2+2a+2 - 15/(a+5)$

Explanation:

First, let's split the top polynomial up into "multiples" of the bottom polynomial. To see what I mean, let's first try to take care of the $3 a^{3}$ $3a^3$ term.

We can see that $3 a^{2} (a + 5) = 3 a^{3} + 15 a^{2}$ $3a^2(a+5) = 3a^3+15a^2$ . So, let's separate this from our polynomial:

$3 a^{3} + 17 a^{2} + 12 a - 5$ $3a^3 + 17a^2 + 12a - 5$

$(3 a^{3} + 15 a^{2}) + 2 a^{2} + 12 a - 5$ $(3a^3+15a^2) + 2a^2+12a-5$

$3 a^{2} (a + 5) + 2 a^{2} + 12 a - 5$ $3a^2(a+5) + 2a^2+12a-5$

See how this "gets rid of" the $3 a^{3}$ $3a^3$ term? Let's do the same for the $2 a^{2}$ $2a^2$ term.

We can see that $2 a (a + 5) = 2 a^{2} + 10 a$ $2a(a+5) = 2a^2 + 10a$ .

$3 a^{2} (a + 5) + (2 a^{2} + 10 a) + 2 a - 5$ $3a^2(a+5) + (2a^2+10a) + 2a-5$

$3 a^{2} (a + 5) + 2 a (a + 5) + 2 a - 5$ $3a^2(a+5) + 2a(a+5)+ 2a-5$

The next highest term to deal with is the $2 a$ $2a$ term.

We can see that $2 (a + 5) = 2 a + 10$ $2(a+5) = 2a+10$ .

$3 a^{2} (a + 5) + 2 a (a + 5) + (2 a + 10) - 15$ $3a^2(a+5) + 2a(a+5)+ (2a+10) -15$

$3 a^{2} (a + 5) + 2 a (a + 5) + 2 (a + 5) - 15$ $3a^2(a+5) + 2a(a+5)+ 2(a+5) - 15$

We can't do anything about the $15$ $15$ , since its degree is smaller than the degree of $a + 5$ $a+5$ .

Finally, let's divide everything by $(a + 5)$ $(a+5)$ . This is why we wrote everything in terms of $a + 5$ $a+5$ , so we can just cancel out the top and the bottom of the fraction at this step.

$\frac{3 a^{3} + 17 a^{2} + 12 a - 5}{a + 5}$ $(3a^3+17a^2+12a-5)/(a+5)$

$= \frac{3 a^{2} (a + 5) + 2 a (a + 5) + 2 (a + 5) - 15}{a + 5}$ $= (3a^2(a+5) + 2a(a+5)+ 2(a+5) - 15)/(a+5)$

$= 3 a^{2} + 2 a + 2 - \frac{15}{a + 5}$ $= 3a^2+2a+2 - 15/(a+5)$

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How do you divide $\frac{3 a^{3} + 17 a^{2} + 12 a - 5}{a + 5}$ $(3a^3+17a^2+12a-5)/(a+5)$ ?

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How do you divide (3a^3+17a^2+12a-5)/(a+5)3a3+17a2+12a−5a+5(3a^3+17a^2+12a-5)/(a+5)?

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How do you divide $\frac{3 a^{3} + 17 a^{2} + 12 a - 5}{a + 5}$ $(3a^3+17a^2+12a-5)/(a+5)$ ?