Question Video: Finding the Remainder Resulting from Dividing Two Polynomials | Nagwa Question Video: Finding the Remainder Resulting from Dividing Two Polynomials | Nagwa

# Question Video: Finding the Remainder Resulting from Dividing Two Polynomials Mathematics

Find the remainder when 4π₯Β² + 4π₯ + 3 is divided by 2π₯ β 3.

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### Video Transcript

Find the remainder when four π₯ squared plus four π₯ plus three is divided by two π₯ minus three.

We set this up as a polynomial long division problem, where weβre dividing four π₯ squared plus four π₯ plus three by two π₯ minus three. The highest-degree term of the divisor is two π₯ and the highest-degree term of the dividend is four π₯ squared.

And weβre asked what is four π₯ squared divided by two π₯. The answer is two π₯. And we put it on top like so, so that it lines up nicely with the π₯ term of the dividend. We take this two π₯, we multiply it by the divisor, two π₯ minus three, and we subtract this product from the dividend.

To make this subtraction easier, we expand the parentheses. And now that the π₯ and π₯ squared terms are lined up, itβs easier to subtract; four π₯ squared minus four π₯ squared is zero π₯ squared or just zero. So we donβt write anything down. Four π₯ minus negative six π₯ is 10π₯, which we do write down, keeping it in the same column as the other π₯ terms. And weβre left with a plus three, from which we donβt subtract anything. So we find that the difference of these two polynomials is 10π₯ plus three.

Weβre not done yet; we can still divide further. Now, we want to divide 10π₯ plus three by two π₯ minus three. The process is very much the same; we ask, what is 10π₯ divided by two π₯? The answer is five. And so our quotient has a plus five, which we put in the same column as the other constant terms.

And now, we need to subtract five times the divisor two π₯ minus three to find whatβs left. As before, we expand these parentheses to make this subtraction easier. On subtracting, we get three minus negative 15, which is 18.

Can we go any further? We canβt divide 18 by two π₯ without entering the world of algebraic fractions. The divisor two π₯ minus three is a degree one polynomial, whereas the constant polynomial 18 has degree zero. The degree of 18 is less than that of our divisor two π₯ minus three, and so we stop dividing.

We find that our quotient is two π₯ plus five, and our remainder is 18. The answer the remainder when four π₯ squared plus four π₯ plus three is divided by two π₯ minus three is therefore 18. And we can make sure that weβve got this right by checking that the dividend four π₯ squared plus four π₯ plus three is equal to the product of the quotient two π₯ plus five and the divisor two π₯ minus three plus the remainder 18.

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