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.
