Video Transcript
Expand the expression five 𝑥 plus
two 𝑦 times 𝑦 minus two 𝑥 squared, giving your answer in its simplest form.
We want to expand and simplify this
expression. We can do this by first expanding
the square of the binomial 𝑦 minus two 𝑥. First of all, we must write 𝑦
minus two 𝑥 squared as the product of 𝑦 minus two 𝑥 and 𝑦 minus two 𝑥. We can then distribute the first
binomial over each term in the second binomial. Then we have 𝑦 times 𝑦 minus two
𝑥 minus two 𝑥 times 𝑦 minus two 𝑥.
We can then distribute each product
to obtain the following products. For the first term, we have 𝑦
times 𝑦 plus negative two 𝑥 times 𝑦. In the second term, we have 𝑦
times negative two 𝑥 plus negative two 𝑥 times negative two 𝑥.
For the next step, we will be using
the product rule for exponents, which states that for any positive integers 𝑚 and
𝑛, we have 𝑥 to the power of 𝑚 times 𝑥 to the power of 𝑛 equals 𝑥 to the power
of 𝑚 plus 𝑛. Using the product rule, we simplify
each term as follows. We can now combine like terms to
get 𝑦 squared minus four 𝑥 𝑦 plus four 𝑥 squared. We can substitute this into the
original expression to obtain five 𝑥 plus two 𝑦 times 𝑦 minus two 𝑥 squared is
equal to five 𝑥 plus two 𝑦 times 𝑦 squared minus four 𝑥𝑦 plus four 𝑥
squared.
Now we will clear part of our work
to make space for the next multiplication. We can now follow the same process
to distribute the factor of five 𝑥 plus two 𝑦 over the trinomial. First, we distribute the binomial
over every term in the trinomial, like so. Second, we distribute each monomial
over the binomials. So we have 𝑦 squared times five 𝑥
plus 𝑦 squared times two 𝑦 plus negative four 𝑥𝑦 times five 𝑥 plus negative
four 𝑥𝑦 times two 𝑦 plus four 𝑥 squared times five 𝑥 plus four 𝑥 squared times
two 𝑦.
We can now simplify each term using
the commutative property of multiplication and the product rule for exponents. So we have five 𝑥𝑦 squared plus
two 𝑦 cubed minus 20𝑥 squared 𝑦 minus eight 𝑥𝑦 squared plus 20𝑥 cubed plus
eight 𝑥 squared 𝑦.
Finally, we rearrange and combine
like terms to get negative three 𝑥𝑦 squared plus two 𝑦 cubed minus 12𝑥 squared
𝑦 plus 20𝑥 cubed. Rearranging the terms to be in
standard form order with respect to 𝑥 gives 20𝑥 cubed minus 12𝑥 squared 𝑦 minus
three 𝑥𝑦 squared plus two 𝑦 cubed.
In conclusion, we have shown that
five 𝑥 plus two 𝑦 times 𝑦 minus two 𝑥 squared simplifies to 20𝑥 cubed minus
12𝑥 squared 𝑦 minus three 𝑥𝑦 squared plus two 𝑦 cubed.
