Question Video: Finding Unknowns by Multiplying Three Binomials | Nagwa Question Video: Finding Unknowns by Multiplying Three Binomials | Nagwa

Question Video: Finding Unknowns by Multiplying Three Binomials Mathematics

If (2π‘₯ βˆ’ 𝑦)(2π‘₯ βˆ’ 5𝑦)(3π‘₯ + 2𝑦) = π‘Žπ‘₯Β³ + 𝑏π‘₯²𝑦 + 𝑐π‘₯𝑦² + 𝑑𝑦³, what are the values of π‘Ž, 𝑏, 𝑐, and 𝑑?

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Video Transcript

If two π‘₯ minus 𝑦 times two π‘₯ minus five 𝑦 times three π‘₯ plus two 𝑦 is equal to π‘Žπ‘₯ cubed plus 𝑏π‘₯ squared 𝑦 plus 𝑐π‘₯𝑦 squared plus 𝑑𝑦 cubed, what are the values of π‘Ž, 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 π‘₯ times two π‘₯ plus two π‘₯ 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 π‘₯ plus negative 𝑦 times negative five 𝑦. Simplifying the results gives four π‘₯ squared minus 10π‘₯𝑦 minus two π‘₯𝑦 plus five 𝑦 squared. Then, we can combine like terms to get four π‘₯ squared minus 12π‘₯𝑦 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 12π‘₯ cubed minus 36π‘₯ squared 𝑦 plus 15π‘₯𝑦 squared. And for the second term, we have eight π‘₯ squared 𝑦 minus 24π‘₯𝑦 squared plus 10𝑦 cubed. Combining the like terms and simplifying gives 12π‘₯ cubed minus 28π‘₯ squared 𝑦 minus nine π‘₯𝑦 squared plus 10𝑦 cubed. We are told this is equal to π‘Žπ‘₯ cubed plus 𝑏π‘₯ squared 𝑦 plus 𝑐π‘₯𝑦 squared plus 𝑑𝑦 cubed. For the polynomials to be equal, their coefficients must be equal. Thus, π‘Ž equals 12, 𝑏 equals negative 28, 𝑐 equals negative nine, and 𝑑 equals 10.

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