Video Transcript
If two 𝑥 minus 𝑦 times two 𝑥
minus five 𝑦 times three 𝑥 plus two 𝑦 is equal to 𝑎𝑥 cubed plus 𝑏𝑥 squared 𝑦
plus 𝑐𝑥𝑦 squared plus 𝑑𝑦 cubed, what are the values of 𝑎, 𝑏, 𝑐, and 𝑑?
On the left-hand side of the
equation, we have the product of three binomials. And on the right-hand side of the
equation, we have a polynomial. We can find a polynomial expression
equivalent to the left-hand side of the equation by expanding the product. Let’s start off by finding the
product of the first two factors. To do this, we can take the product
of each pair of terms from the first parentheses with the second parentheses and add
the results.
So, we can distribute the first
term from the first binomial through both terms in the second binomial, resulting in
two 𝑥 times two 𝑥 plus two 𝑥 times negative five 𝑦. Then, we can distribute the second
term from the first binomial through both terms in the second binomial, resulting in
negative 𝑦 times two 𝑥 plus negative 𝑦 times negative five 𝑦. Simplifying the results gives four
𝑥 squared minus 10𝑥𝑦 minus two 𝑥𝑦 plus five 𝑦 squared. Then, we can combine like terms to
get four 𝑥 squared minus 12𝑥𝑦 plus five 𝑦 squared. Now we can substitute this
expression into our product of three binomials. Then, we can expand the product by
distributing the trinomial over each term in the binomial.
For the first term, we have 12𝑥
cubed minus 36𝑥 squared 𝑦 plus 15𝑥𝑦 squared. And for the second term, we have
eight 𝑥 squared 𝑦 minus 24𝑥𝑦 squared plus 10𝑦 cubed. Combining the like terms and
simplifying gives 12𝑥 cubed minus 28𝑥 squared 𝑦 minus nine 𝑥𝑦 squared plus 10𝑦
cubed. We are told this is equal to 𝑎𝑥
cubed plus 𝑏𝑥 squared 𝑦 plus 𝑐𝑥𝑦 squared plus 𝑑𝑦 cubed. For the polynomials to be equal,
their coefficients must be equal. Thus, 𝑎 equals 12, 𝑏 equals
negative 28, 𝑐 equals negative nine, and 𝑑 equals 10.
