Video Transcript
We want to factor 32π₯ to the fourth plus 100π₯ cubed plus 35π₯ squared minus 63π₯ into two factors. Given that one of these factors is four π₯ squared plus nine π₯, 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 π₯ squared to make it look like 32π₯ to the fourth. That would be eight π₯ squared.
And since itβs an π₯ squared term, weβll put it above the π₯ squared term. And now we distribute and we subtract them and get 28π₯ cubed.
And now we repeat the process again. What do we multiply four π₯ squared by in order to look like 28π₯ cubed? That would be seven π₯. So we distribute, but we also need to bring down the 35π₯ squared. And when we subtract, we get negative 28π₯ squared.
Repeating the process again, we would multiply four π₯ squared by negative 7 to look like negative 28π₯ 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 63π₯, and subtract, and we get zero.
So our remainder is zero. Thereβs nothing left over. And since we know that these two factors multiply to equal this polynomial, we shouldnβt get a remainder anyway. So our final answer is eight π₯ squared plus seven π₯ minus seven.