Find 𝐴𝐵 given that 𝐴 equals eight 𝑥 plus two and 𝐵 equals five 𝑥 minus one.
𝐴𝐵 will be found by multiplying the expression 𝐴 times the expression for 𝐵, eight 𝑥 plus two times five 𝑥 minus one. We’ll need to expand this multiplication. We multiply the first two terms, eight 𝑥 times five 𝑥, then the outer two terms, eight 𝑥 times negative one. Make sure that you bring that negative with the one value. It’s not eight 𝑥 times one. It’s eight 𝑥 times negative one.
For now, we multiply the insides, two times five 𝑥, and then the lasts, two times negative one. Again make sure you carry that negative one down, two times negative one. And then, we’ll add all four of these terms.
First, we’ll multiply the coefficients. Eight times five equals 40. We have the 𝑥-variable. And we add the two exponents. 𝑥 times 𝑥 equals 𝑥 squared, 𝑥 to the one plus one power. Eight 𝑥 times five 𝑥 equals 40𝑥 squared. This time, we’ll multiply eight by negative one, which is negative eight. And we have one 𝑥-variable. The same thing for two times five 𝑥. We’ll multiply two times five, which is 10, and bring down the 𝑥. For the final term, two times negative one equals negative two. And we’re trying to combine these terms together.
We now need to look and see if any of these terms can be combined. You can combine terms who have the same variable and the same exponent. Negative eight 𝑥 can be added to 10𝑥. You can imagine negative eight 𝑥 as eight negative 𝑥 terms. And you can imagine 10𝑥 as 10 positive 𝑥 terms. When you add a negative and a positive together, they cancel out. And that means negative eight 𝑥 plus 10𝑥 leaves us with two 𝑥, two positive 𝑥 values. We bring down the 40𝑥 squared, which hasn’t changed, and the negative two, which hasn’t changed.
And we can say that 𝐴𝐵 equals 40𝑥 squared plus two 𝑥 minus two.