Video Transcript
Find the value of 𝑎, where two 𝑥 squared plus three 𝑥 plus 𝑎 divided by 𝑥 plus one has no remainder.
We first recall that we say a polynomial is divisible by another polynomial if their division has a remainder of zero. This means we can apply polynomial long division to the given polynomials. And we know that the remainder must be zero.
To apply polynomial long division, we first need to divide the leading terms. We find the first term of our quotient by dividing two 𝑥 squared by 𝑥, which equals two 𝑥. So we add two 𝑥 to the quotient. Then, we subtract two 𝑥 times the divisor from the dividend. Specifically, that is two 𝑥 squared plus two 𝑥. We must be careful to subtract both terms, not just the first term. It may be helpful to first distribute the negative then combine like terms. This means subtracting two 𝑥 squared and subtracting two 𝑥 as follows.
The quadratic terms cancel, leaving us with 𝑥 plus 𝑎. We recall that as long as the degree of the new dividend is greater than or equal to the degree of the divisor, we can perform another round of division. In this round, we divide the leading term from the new dividend by the leading term of the divisor. Since they are both 𝑥, we get one. So we add one to the quotient. Then, we subtract one times the divisor, which is 𝑥 plus one.
Once again, we must carefully subtract both terms as follows, subtracting 𝑥 from 𝑥 and subtracting one from 𝑎. Finally, we are left with a remainder of 𝑎 minus one. Since we are told the division is exact, we know that there should be no remainder. Hence, 𝑎 minus one must equal zero. We can solve this equation to see that 𝑎 is equal to one. We have found the value of 𝑎, where two 𝑥 squared plus three 𝑥 plus 𝑎 divided by 𝑥 plus one has no remainder.
