We're asked to simplify
by using the distributive property and combining like terms.
The distributive property is shown below,
where we have a times the quantity b plus c
equals a times b plus a times c,
where we say we distributed the a,
what is often referred to as multiplication across addition.
Again, a times the quantity b plus c
is equal to a times b
plus a times c.
Looking at the first example,
we have negative five times the quantity five x plus three.
We begin by distributing the negative five,
and because we have a sum of two terms in the parentheses,
we will have two products.
This is equal to negative five times five x
plus negative five times three.
Now we'll determine the products.
Negative five times five x equals negative 25 x.
Then we have plus negative five times three
is equal to negative 15.
Instead of writing plus negative 15,
it's more common just to write minus 15.
We enter negative 25x minus 15.
Next, we have negative eight times the quantity
negative six w squared minus nine w plus two.
Notice, inside the parentheses,
we have a sum or difference of three terms,
which means when we distribute, we will have three products.
One,
two,
and three.
This is equal to negative eight
times negative six w squared,
and then because we have a minus here,
we have minus negative eight times nine w,
and then plus negative eight times two.
Simplifying, negative eight times negative six w squared
is equal to positive 48w squared,
and then we have minus negative eight times nine w,
is going to be minus negative 72w.
Subtracting a negative is the same as adding a positive.
This simplifies to plus 72w,
and then we have plus negative eight times two,
which is plus negative 16, or just minus 16.
We enter 48w squared
plus 72w
minus 16.
Next we have negative, or the opposite of,
negative t squared plus six t minus seven.
To simplify using distribution,
we can include a factor of one here,
and then think of distributing negative one,
so we'll have one,
two,
three products when we distribute negative one.
This is equal to negative one times negative t squared
plus negative one times six t,
and then minus negative one times seven.
Now we determine the products.
Negative one times negative t squared
is equal to positive t squared,
and then we have plus negative one times six t,
which would be plus negative six t or minus six t,
and then we have minus negative one times seven,
which would be minus negative seven, or plus seven.
This simplifies to t squared minus six t plus seven.
Now, if we go back and think of this as
the opposite of negative t squared plus six t minus seven,
we would change the sign
of each term inside the parantheses,
which, notice, is the result.
The result is positive t squared minus six t plus seven.
Now, for the last example, we have 5/8 times the quantity
1/3x plus 1/4,
we distribute 5/8,
which means we'll have two products.
One,
and two.
This is equal to 5/8
times 1/3x
plus 5/8
times 1/4.
Before multiplying, notice how
there are no common factors on than one
between the numerator and denominator,
and therefore we can multiply,
knowing the product will not simplify.
5/8 times 1/3x is equal to 5/24x,
and then we have plus 5/8 times 1/4
is equal to 5/32,
so we enter 5/24x
plus 5/32.
I hope you found this helpful.