# Video: Writing Rational Expressions in Different Forms by Comparing Coefficients

Bethani Gasparine

Write (3𝑥³ + 4𝑥² + 5𝑥 + 10)/(𝑥 + 5) in the form of 𝑞(𝑥) + (𝑟(𝑥))/(𝑑(𝑥)).

01:25

### Video Transcript

Write three 𝑥 cubed plus four 𝑥 squared plus five 𝑥 plus ten divided by 𝑥 plus five in the form 𝑞 of 𝑥 plus 𝑟 of 𝑥 divided by 𝑑 of 𝑥.

Essentially we will be dividing these creating a new polynomial and then we will be adding the remainder to it. When dividing, the first thing we need to do is decide what do we multiply to 𝑥 so that it’s equal to three 𝑥 cubed. We will need to multiply 𝑥 by three 𝑥 squared for it to be equal to three 𝑥 cubed, so we put it above the 𝑥 squared term. And now we distribute, and now we subtract, and we get negative 11𝑥 squared.

So what do we multiply to 𝑥 so that it’s equal to negative 11𝑥 squared? That would be negative 11𝑥. Distribute. And we also need to bring down the five 𝑥. And now we can subtract, and we get 60𝑥. So how do we get 𝑥 to look like 60𝑥? We multiply by 60. And now we distribute. We need to bring down our 10, and now we subtract, and we are left with negative 290. That is our remainder. And we put our reminder over what we actually divided by. So the final answer is three 𝑥 squared minus 11𝑥 plus 60 minus 290 divided by 𝑥 plus five.

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