Question Video: Multiplying a Binomial by a Trinomial | Nagwa Question Video: Multiplying a Binomial by a Trinomial | Nagwa

Question Video: Multiplying a Binomial by a Trinomial Mathematics • First Year of Preparatory School

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Simplify the expression (𝑥 + 2)(𝑥² − 5𝑥 + 3).

02:27

Video Transcript

Simplify the expression 𝑥 plus two multiplied by 𝑥 squared minus five 𝑥 plus three.

The expression we’ve been asked to simplify is the product of a binomial, 𝑥 plus two, and a trinomial, 𝑥 squared minus five 𝑥 plus three. To simplify this expression means we need to distribute the parentheses and then combine any like terms. To answer this question, we need to recall the distributive property of multiplication over addition. This tells us that if we are multiplying 𝑎 by the sum of 𝑏 and 𝑐, this is equivalent to 𝑎 multiplied by 𝑏 plus 𝑎 multiplied by 𝑐.

To apply this property, we could distribute the factor of 𝑥 plus two over the addition and subtraction in the trinomial, but it is easier to distribute the other way round. We may find it helpful to write the two factors in the opposite order, which we know is valid due to multiplication being commutative. Then, distributing the factor of 𝑥 squared minus five 𝑥 plus three over the binomial gives 𝑥 multiplied by 𝑥 squared minus five 𝑥 plus three plus two multiplied by 𝑥 squared minus five 𝑥 plus three.

Each part of the expression on the right-hand side of the equation is now the product of a monomial and a trinomial. We can now apply the distributive property again by distributing each monomial over the trinomial. Distributing 𝑥 over the trinomial and recalling that when we multiply powers of the same base, we add the exponents gives 𝑥 cubed minus five 𝑥 squared plus three 𝑥. Then distributing the two over the trinomial gives two 𝑥 squared minus 10𝑥 plus six. Finally, we need to simplify by grouping like terms. Doing so gives 𝑥 cubed minus three 𝑥 squared minus seven 𝑥 plus six.

So, using the distributive property twice, we’ve shown that the simplified form of the given expression is 𝑥 cubed minus three 𝑥 squared minus seven 𝑥 plus six.

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