Find the remainder when four 𝑥 squared plus four 𝑥 plus three is divided by two 𝑥 minus three.
We set this up as a polynomial long division problem, where we’re dividing four 𝑥 squared plus four 𝑥 plus three by two 𝑥 minus three. The highest-degree term of the divisor is two 𝑥 and the highest-degree term of the dividend is four 𝑥 squared.
And we’re asked what is four 𝑥 squared divided by two 𝑥. The answer is two 𝑥. And we put it on top like so, so that it lines up nicely with the 𝑥 term of the dividend. We take this two 𝑥, we multiply it by the divisor, two 𝑥 minus three, and we subtract this product from the dividend.
To make this subtraction easier, we expand the parentheses. And now that the 𝑥 and 𝑥 squared terms are lined up, it’s easier to subtract; four 𝑥 squared minus four 𝑥 squared is zero 𝑥 squared or just zero. So we don’t write anything down. Four 𝑥 minus negative six 𝑥 is 10𝑥, which we do write down, keeping it in the same column as the other 𝑥 terms. And we’re left with a plus three, from which we don’t subtract anything. So we find that the difference of these two polynomials is 10𝑥 plus three.
We’re not done yet; we can still divide further. Now, we want to divide 10𝑥 plus three by two 𝑥 minus three. The process is very much the same; we ask, what is 10𝑥 divided by two 𝑥? 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 𝑥 minus three to find what’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 can’t divide 18 by two 𝑥 without entering the world of algebraic fractions. The divisor two 𝑥 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 𝑥 minus three, and so we stop dividing.
We find that our quotient is two 𝑥 plus five, and our remainder is 18. The answer the remainder when four 𝑥 squared plus four 𝑥 plus three is divided by two 𝑥 minus three is therefore 18. And we can make sure that we’ve got this right by checking that the dividend four 𝑥 squared plus four 𝑥 plus three is equal to the product of the quotient two 𝑥 plus five and the divisor two 𝑥 minus three plus the remainder 18.