We want to factor 32𝑥 to the fourth plus 100𝑥 cubed plus 35𝑥 squared minus 63𝑥 into two factors. Given that one of these factors is four 𝑥 squared plus nine 𝑥, what is the other?
When we factor a polynomial, the factors multiply to equal the polynomial. So if we know one of the factors, we can divide and solve for the other factor.
So in order to divide, we need to figure out what do we multiply to four 𝑥 squared to make it look like 32𝑥 to the fourth. That would be eight 𝑥 squared.
And since it’s an 𝑥 squared term, we’ll put it above the 𝑥 squared term. And now we distribute and we subtract them and get 28𝑥 cubed.
And now we repeat the process again. What do we multiply four 𝑥 squared by in order to look like 28𝑥 cubed? That would be seven 𝑥. So we distribute, but we also need to bring down the 35𝑥 squared. And when we subtract, we get negative 28𝑥 squared.
Repeating the process again, we would multiply four 𝑥 squared by negative 7 to look like negative 28𝑥 squared. And negative seven is a constant term, so we would put it above the constant term, which would really be a zero. Distribute, bring down the negative 63𝑥, and subtract, and we get zero.
So our remainder is zero. There’s nothing left over. And since we know that these two factors multiply to equal this polynomial, we shouldn’t get a remainder anyway. So our final answer is eight 𝑥 squared plus seven 𝑥 minus seven.