Video Transcript
Find the quotient of two 𝑥 squared plus three 𝑥 plus one divided by 𝑥 plus one.
We can find an expression for two 𝑥 squared plus three 𝑥 plus one divided by 𝑥 plus one using long division. The first step is to find the quotient of the leading terms of the dividend and the divisor. We note that two 𝑥 squared divided by 𝑥 is equal to two 𝑥. So we write two 𝑥 in the quotient.
Next, we need to subtract two 𝑥 times the divisor from the dividend. So, first, we find that two 𝑥 times the divisor, 𝑥 plus one, equals two 𝑥 squared plus two 𝑥. This will be subtracted from the dividend. So we line up the like terms in columns and then subtract the expression we found by multiplying two 𝑥 and the divisor.
We must be careful to subtract each term, not just the first term. This is why we use parentheses. It may also help to think of this step as distributing a negative through the polynomial and then combining the like terms. This means we subtract two 𝑥 squared and subtract two 𝑥 from the dividend, like so. We see that the quadratic terms cancel, and we are left with 𝑥 plus one. We call 𝑥 plus one the remainder. If the remainder is zero, we can say that the dividend is divisible by the divisor. However, as long as the remainder has a degree 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 two 𝑥 squared plus three 𝑥 plus one divided by 𝑥 plus one is equal to two 𝑥 plus 𝑥 plus one over 𝑥 plus one, where 𝑥 plus one in the numerator is our remainder and 𝑥 plus one in the denominator is our divisor. Since the degree of the remainder and the degree of the divisor are equal, we have another round of division to perform. This means the remainder, 𝑥 plus one, is the new dividend. In this case, the leading terms of the dividend and divisor are equal. So, when we take their quotient, we get positive one. So we add one to the quotient. Next, we subtract one times the divisor from the new dividend, leaving no remainder.
Therefore, the quotient of two 𝑥 squared plus three 𝑥 plus one divided by 𝑥 plus one is equal to two 𝑥 plus one. Having found the quotient with a zero remainder, we can also say that two 𝑥 squared plus three 𝑥 plus one is divisible by 𝑥 plus one.
