Video Transcript
Find π΄π΅ given π΄ equals five π₯ cubed minus three π₯ and π΅ equals negative six π₯ squared plus three π₯.
π΄π΅ would be multiplying the expression of π΄ and the expression of π΅. We need to multiply five π₯ cubed minus three π₯ times negative six π₯ squared plus three π₯. We do that by foiling. Multiplying the firsts, the outers, the inners, and the lasts. The first, five π₯ cubed times negative six π₯ squared. And then, five π₯ cubed times three π₯. So weβll add five π₯ cubed times three π₯.
We need to be careful here. Weβre multiplying negative three π₯ by negative six π₯ squared. That means weβll be adding negative three π₯ times negative six π₯ squared and then negative three π₯ times positive three π₯. So weβll add negative three π₯ times positive three π₯. Five times negative six equals negative 30. We bring down the variable and then add the two exponents. Three plus two is five. Five π₯ cubed times negative six π₯ squared equals negative 30π₯ to the fifth.
Follow the same procedure. Multiply the coefficients. Five times three equals 15. Bring down the π₯. Three π₯ is the same thing as three times π₯ to the first power. And three plus one is four. Five π₯ cubed times three π₯ equals 15π₯ to the fourth.
The next term, weβll multiply negative three times negative six, which is positive 18. And then, weβll add the exponents of the two π₯s. One plus two is three. The last term has negative three times positive three, which is negative nine. Both of these π₯-values have an exponent of one. π₯ times π₯ is the same thing as π₯ to the one plus one power. Itβs π₯ squared.
Weβre adding all these terms together, negative 30π₯ to the fifth plus 15π₯ to the fourth plus 18π₯ cubed. For the last term, weβre adding a negative. And we can simplify that by saying subtracting nine π₯ squared.
π΄π΅ equals negative 30π₯ to the fifth plus 15π₯ to the fourth plus 18π₯ cubed minus nine π₯ squared.
