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