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Multiplying polynomials:
- use the distributive property and the properties of exponents
- here the distributive property can be used to distribute one term or
multiple terms
- after multiplying be sure to combine all like terms
Example 1 : Multiply the polynomials and express your answers as
simplified polynomials.
a. ( 2 ๐ฅ + 5 )( 3 ๐ฅ โ 7 ) b. ( 3 ๐ฅ โ 4 )( 3 ๐ฅ + 4 )
b.
F O I L F O I L
c. (๐ฅ
4
2
4
2
) d. ( 3 ๐ฅ + 5 )( 2 ๐ฅ
2
d.
F O I L
(๐ฅ
4
)(๐ฅ
4
) + (๐ฅ
4
)(โ 3 ๐ฆ
2
) + ( 3 ๐ฆ
2
)(๐ฅ
4
) + ( 3 ๐ฆ
2
)(โ 3 ๐ฆ
2
)
8
4
2
4
2
4
๐
๐
e.
2
2
Since this is a trinomial times a trinomial I canโt use FOIL, so instead
Iโll simply write each term from the first trinomial times the entire
second trinomial.
2
2
2
2
4
3
2
3
2
2
๐
f. ( 5 โ ๐ฅ)(๐ฅ + 5 )(๐ฅ โ 5 )
Since this is a binomial times a binomial times another binomial, I
can only use FOIL to multiply two of the binomials. It makes no
difference which two I choose to multiply first, so Iโll just work from
left to right and multiply the first two binomials first:
2
2
Now that Iโve multiplied the first two binomials and simplified
completely, I can take that product and multiply by the third
binomial. And since I have two binomials, I can once again use
FOIL.
2
๐
๐
c. โ 5
3
3
2
d.
2
2
3
3
3
3
2
2
2
2
2
2
4
๐
๐
Once again, when you have a
product containing more than
two factors, it makes no
difference which two factors you
choose to multiply first; the order
is irrelevant. In this example Iโll
re-arrange the factors so I can
multiply ( 3 โ ๐ฅ) and ( 3 + ๐ฅ).
The next example contains problem parts that are similar to some past
exam problems that students have had trouble with. As I go through
Example 3, please be sure you are paying attention and working through
those problems with me, and please ask questions if youโre unsure about
anything.
Example 3 : Multiply the polynomials and express your answers as
simplified polynomials.
a. 5 ๐ฅ
2
b. ๐ฅ
4
2
2
2
4
2
2
4
2
2
4
2
2
2
4
2
2
4
2
2
4
2
2
4
2
2
๐
๐
๐
๐
you need assistance understanding how to simplify these types of
expressions, please let me know.
The next example has two expressions that are not polynomials because
the exponents are fractions ( โ
1
2
) instead of nonnegative integers.
However we can still multiply and combine like terms the same way we
have with polynomials.
Example 4: Multiply the following expressions and simplify your answer
as much as possible. Keep in mind that while these expressions are not
polynomials, but they can still be multiplied using the same procedure.
a. ( โ
2
b. ( โ
2
1
0
c. ๐
Answers to Examples:
1a 6 ๐ฅ
2
- ๐ฅ โ 35 ; 1 b. 9 ๐ฅ
2
โ 16 ; 1 c ๐ฅ
8
4
1 d. 6 ๐ฅ
3
2
- 30 ๐ฅ โ 25 ; 1 e. ๐ฅ
4
3
2
; 2 a. ๐ฅ
2
โ 2 ๐ฅ + 1 ; 2 b. 64 ๐ฅ
3
2
2 c. โ 5 ๐ฅ
6
3
3
6
; 2 d. ๐ฅ
4
2
3a. โ๐ฅ
2
โ 8 ๐ฅ + 6 ; 3b. โ๐ฅ
2
2
2
2
; 3c. 11 ๐ฅ
2
3d. ๐ฅ
3
2
2
3
2
2
4a. 2 ๐ฆ + 2 โ
๐ฅ๐ฆ ; 4b. ๐ฅ โ 2
โ
