Example 2
(3x
–
4)(2x + 5)
= (3x)(2x) + (3x)(5) + (-4)(2x) + (-4)(5)
= 6x
2
+ 15x
–
8x
–
20
= 6x
2
+ 7x
–
20
To verify, let x = 2
(3x
–
4)(2x + 5)
=
6x
2
+ 7x
–
20
(3(2)
–
4)(2(2) + 5)
6(2)
2
+ 7(2)
–
20
(6
–
4)(4 + 5)
6(4) + 14
–
20
(2)(9)
24
–
6
18
18
Since both sides equal 18, (3x
–
4)(2x + 5) = 6x
2
+ 7x
–
20
The algebraic method is most widely used and will be the
most efficient
method when
determining more complex products such as the one that follows below.
N.B
.
We can use
any
value in our check!
N.B.
Notice the
format
of a verification!
There is
a
?
over the equal sign and a
line
separating the
two sides of the equation.
This is
necessary
until
we have shown the two sides to be equal!
?
?
Notice order of
operations!
We
square 2 then multiply
by 6.

10C
–
P1-2 DETAILED SOLUTIONS
Page 28 of 47
Example
(3x
2
+ 2)(x
2
–
2x)
= (3x
2
)(x
2
) + (3x
2
)(-2x) + (2)(x
2
) + (2)(-2x)
= 3x
4
–
6x
3
+ 2x
2
–
4x
Actually, we can use algebra and the distributive property to multiply
any
number of terms by
any
number of terms.
Let‟s show this by multiplying a trinomial times a trinomial as in the example that follows.
Example
(x
2
+ 2x + 3)(2x
2
–
3x + 1)
First
term in first trinomial is multiplied by
each
term in second trinomial
= (x
2
)(2x
2
) + (x
2
)(-3x) + (x
2
)(1)
= 2x
4
–
3x
3
+ x
2
result 1
Second
term in first trinomial is multiplied by
each
term in second trinomial
= (2x)(2x
2
) + (2x)(-3x) + (2x)(1)
= 4x
3
–
6x
2
+ 2x
result 2
Third
term in first trinomial is multiplied by
each
term in second trinomial
= (3)(2x
2
) + (3)(-3x) + (3)(1)
= 6x
2
–
9x + 3
result 3
The final product is
result 1 + result 2 + result 3
.
2x
4
–
3x
3
+ x
2
+ 4x
3
–
6x
2
+ 2x + 6x
2
–
9x + 3
= 2x
4
–
3x
3
+ 4x
3
+ x
2
–
6x
2
+ 6x
2
+ 2x
–
9x + 3
= 2x
4
+ x
3
+ x
2
–
7x + 3
When multiplying polynomials, it is important to remember that each term in
the first polynomial is multiplied by each term in the second polynomial.
Simplify
the expression by
combining
like
terms.

10C
–
P1-2 DETAILED SOLUTIONS
Page 29 of 47
Let‟s verify using x =
-1
It is necessary to place -1 in parentheses as shown below!
(x
2
+ 2x + 3)(2x
2
–
3x + 1)
=
2x
4
+ x
3
+ x
2
–
7x + 3
((-1)
2
+ 2(-1) + 3)(2(-1)
2
–
3(-1) + 1)
2(-1)
4
+ (-1)
3
+ (-1)
2
–
7(-1) + 3
(1 + -2 + 3)(2 + 3 + 1)
2 + -1 + 1 + 7 + 3
(2)(6)
13 + -1
12
12
Since both sides worked out to be 12,
(x
2
+ 2x + 3)(2x
2
–
3x + 1) = 2x
4
+ x
3
+ x
2
–
7x + 3
Now that you understand how to multiply polynomials, you will be asked to
demonstrate
your understanding in a variety of ways.
Carefully read the following examples.
Example 1
Suzie‟s solution to a multiplication question appears below.
Identify and explain
the error that she made.
(3x
–
2)(2x + 5)
= (3x)(2x) + (-2)(5)
= 6x
2
–
10
Example 2
Explain to Justin the error that he made when he multiplied the following
binomials.
(4x
–
1)(2x
–
3)
= (4x)(2x) + (4x)(-3)
= 8x
2
–
12x
Refer to
Foundations and Pre-Calculus Mathematics
10
.
On page 167, complete Question 13
To summarize, when multiplying polynomials it is necessary to
Multiply
each
term in the first polynomial by
each
term in the second polynomial.

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- Winter '19
- jane smith
- Math, Algebra, Polynomials, Coefficient, constant term