Video Transcript
What does the expression 𝑥 plus
two times 𝑥 minus seven times 𝑥 plus four simplify to?
We are asked to expand and simplify
the product of three binomials. To do this, we first note that
multiplication is associative. Thus, we can evaluate the product
of these binomials by first multiplying two of the binomials together.
Let’s expand and simplify 𝑥 plus
two times 𝑥 minus seven. We can do this by using the
horizontal method. We first distribute the binomial 𝑥
plus two through the terms in the second set of parentheses. So, we have 𝑥 times 𝑥 plus two
minus seven times 𝑥 plus two. To simplify further, we need to
distribute each factor over the parentheses. So, we take 𝑥 times 𝑥 plus 𝑥
times two minus seven times 𝑥 minus seven times two. This simplifies to 𝑥 squared plus
two 𝑥 minus seven 𝑥 minus 14.
We can then simplify by combining
the like terms, two 𝑥 and negative seven 𝑥. Two minus seven is negative
five. Therefore, two 𝑥 minus seven 𝑥 is
negative five 𝑥. So, the simplified product of 𝑥
plus two and 𝑥 minus seven is 𝑥 squared minus five 𝑥 minus 14. Thus, 𝑥 plus two times 𝑥 minus
seven times 𝑥 plus four equals 𝑥 squared minus five 𝑥 minus 14 times 𝑥 plus
four.
We will now clear some space for
the next round of multiplication. We now distribute the factor of 𝑥
plus four over the trinomial. This gives us 𝑥 squared times 𝑥
plus four minus five 𝑥 times 𝑥 plus four minus 14 times 𝑥 plus four. To simplify further, we need to
distribute each factor over the parentheses. We can do this by recalling the
product rule for exponents. This tells us that 𝑥 to the power
of 𝑚 times 𝑥 to the power of 𝑛 is equal to 𝑥 to the power of 𝑚 plus 𝑛.
And remember, whenever we’re
talking about monomials, we’re talking about our powers being nonnegative integer
values. So, in particular, the product rule
for monomials only counts when 𝑚 and 𝑛 are nonnegative integers, although this
formula is true for other values of 𝑚 and 𝑛.
We continue by evaluating the
product for each term separately. We get 𝑥 squared times 𝑥 plus
four equals 𝑥 cubed plus four 𝑥 squared. Negative five 𝑥 times 𝑥 plus four
equals negative five 𝑥 squared minus 20𝑥. And negative 14 times 𝑥 plus four
equals negative 14𝑥 minus 56. Substituting these into our
expression gives 𝑥 cubed plus four 𝑥 squared minus five 𝑥 squared minus 20𝑥
minus 14𝑥 minus 56. We can now simplify by combining
like terms as follows, which equals 𝑥 cubed minus 𝑥 squared minus 34𝑥 minus
56.
In conclusion, we have shown that
𝑥 plus two times 𝑥 minus seven times 𝑥 plus four simplifies to 𝑥 cubed minus 𝑥
squared minus 34𝑥 minus 56.
