Question Video: Dividing Polynomials Using Polynomial Long Division Mathematics

Video Transcript

Find the quotient of 𝑥 squared plus two 𝑥 minus three divided by 𝑥 minus one.

We can find an expression for 𝑥 squared plus two 𝑥 minus three divided by 𝑥 minus one by using long division. The first step is to find the quotient of the leading terms of the dividend and the divisor. We note that 𝑥 squared divided by 𝑥 is equal to 𝑥. So we write 𝑥 in the quotient.

Next, we need to subtract 𝑥 times the divisor from the dividend. After distributing 𝑥 into the parentheses, we have 𝑥 squared minus 𝑥. This will be subtracted from the dividend. So we line up the like terms in columns and then subtract the expression we just found by multiplying 𝑥 with the divisor. We must be very careful to subtract each term, not just the first term. This is why we use parentheses. It may be helpful to think of this step as distributing a negative through the polynomial and then combining like terms. This means we actually subtract 𝑥 squared and add 𝑥 to the dividend.

The result of combining like terms is three 𝑥 minus three. The quadratic terms cancel out. We call three 𝑥 minus three the remainder. If the remainder is zero, we can say that the dividend is divisible by the divisor. However, as long as the degree of the remainder is greater than or equal to the degree of the divisor, then we can still perform another round of division.

Let’s pause to clarify what we have determined so far. We have shown that 𝑥 squared plus two 𝑥 minus three divided by 𝑥 minus one is equal to 𝑥 plus three 𝑥 minus three over 𝑥 minus one, where three 𝑥 minus three is our remainder and 𝑥 minus one is our divisor. Since the degree of the remainder and the divisor are equal, we have another round of division to perform. Therefore, the remainder, three 𝑥 minus three, is the new dividend.

In this round, the quotient of the leading terms, three 𝑥 and 𝑥, is equal to three. So we add three to the quotient. Then, we compute three times the divisor, 𝑥 minus one, to get three 𝑥 minus three. We then subtract this expression from the new dividend. We line up the like terms in columns. And we must be very careful to subtract each term, not just the first term. It may be easier to think of this step as distributing a negative through the polynomial and then combining like terms. This looks like subtracting three 𝑥 and adding three to the new dividend, leaving us with a remainder of zero. Therefore, 𝑥 squared plus two 𝑥 minus three is divisible by 𝑥 minus one, and their quotient is 𝑥 plus three.

An alternative way of completing this division would have been to factor the highest common factor, three, from the new dividend. Then, when three times 𝑥 minus one is divided by 𝑥 minus one, the second term of the final quotient simplifies to three. Whichever method we use for the last part of the division, we still end up with the correct quotient of 𝑥 plus three.