Video: Dividing Polynomials

Find the quotient when 6π‘₯⁴ + 11π‘₯Β³ βˆ’ 43π‘₯Β² βˆ’ 40π‘₯ is divided by π‘₯Β² + π‘₯ βˆ’ 8.

02:30

Video Transcript

Find the quotient when six π‘₯ to the fourth plus 11π‘₯ cubed minus 43π‘₯ squared minus 40π‘₯ is divided by π‘₯ squared plus π‘₯ minus eight.

So we are finding a quotient, which means that we’re dividing. And it says we’re taking six π‘₯ to the fourth plus 11π‘₯ cubed minus 43π‘₯ squared minus 40π‘₯ and dividing it by π‘₯ squared plus π‘₯ minus eight. So to do this, we look at the first terms. And we think, how many π‘₯ squareds go into six π‘₯ to the fourth? Or what do we multiply π‘₯ squared by to make it be six π‘₯ to the fourth?

We would need to multiply π‘₯ squared by six π‘₯ squared. That would give us six π‘₯ to the fourth. And we put the π‘₯ squared term above the π‘₯ squared term here. This keeps us organised. Now, we need to begin multiplying six π‘₯ squared to π‘₯ squared plus π‘₯ minus eight. So six π‘₯ squared times π‘₯ squared is six π‘₯ to the fourth power.

Now, we take six π‘₯ squared times π‘₯, which is six π‘₯ cubed. And notice we’re putting the π‘₯ to the fourth powers and π‘₯ cubes underneath each other. And now we take six π‘₯ squared times negative eight, giving us negative 48π‘₯ squared. And we will be subtracting. These cancel. And then, 11π‘₯ cubed minus six π‘₯ cubed will be five π‘₯ cubed.

Now, we have negative 43π‘₯ squared minus negative 48π‘₯ squared. So really, we’re taking negative 43π‘₯ squared plus 48π‘₯ squared, which is positive five π‘₯ squared. And we will need to bring down the minus 40π‘₯.

So now we’ll repeat this process again. What do we multiply to π‘₯ squared to make it be five π‘₯ cubed? We’ll need to multiply π‘₯ squared by five π‘₯. And now we distribute. So five π‘₯ times π‘₯ squared is five π‘₯ cubed. Five π‘₯ times π‘₯ is five π‘₯ squared. And five π‘₯ times negative eight is negative 40π‘₯. And remember, we’re subtracting.

So five π‘₯ cubed minus five π‘₯ cubed cancels. Five π‘₯ squared minus five π‘₯ squared cancels. And then negative 40π‘₯ minus negative 40π‘₯ is negative 40π‘₯ plus 40π‘₯. So those cancel as well, which means there’s nothing left. There’s no remainder. So essentially, this divided evenly.

And our quotient, our answer will be six π‘₯ squared plus five π‘₯.

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