2.01
Products of Polynomials
from Basic
Algebra: One Step at
a Time © 2002-2011
P.
121–132
Dr. Robert J. Rapalje
Seminole State College of Florida
Sanford, FL 32773
ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE
END OF THIS PAGE
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Recall
from previous lessons that when algebraic expressions are added (or
subtracted) they are called
terms, while expressions that are
multiplied are called factors. An algebraic expression that
contains only one term is called a
monomial. If the
expression has two terms, it is called a
binomial,
and if there are three terms, it is a
trinomial.
A
polynomial
is an algebraic expression consisting of one or more
terms. A polynomial may consist of numbers and variables, where the
numerical part of a given term is called the
coefficient. If there
is only one variable in the polynomial, such as x, then it is
called a
polynomial in x. The
degree (or
order)
of
a
polynomial in one variable is the highest exponent of the variable. If
there is more than one variable in the polynomial, then the degree
(or order) is the highest "sum of the exponents" of the variables
of a given term.
Frequently polynomials can be simplified by combining like terms;
sometimes they can be factored. Polynomials can be added,
subtracted, multiplied (expanded), or divided. Since addition
and
subtraction of polynomials is little more than combining like terms, and
division of polynomials is saved for Chapter 2, this section will involve
only the multiplication (expansion) of polynomial expressions. The
next section is the factoring of polynomial expressions, followed
immediately by solving quadratic equations by factoring. Notice
that polynomial expressions are not equations, and therefore
cannot be "solved." This chapter involves only polynomial
expressions.
This
explanation will begin with a review of products (monomial times various
polynomials) from Chapter 1. Then we will do a binomial times
binomial, binomial times trinomial, and trinomial times trinomial.
The basic property that underlies these products is the distributive
property for multiplication (products) over addition (two or
more terms).
121
Monomial Times a Polynomial
A
monomial may be multiplied times a polynomial by simply using the
distributive property.
EXAMPLE 1.
Monomial times monomial
a) (3x)•(4
x) b) (3 x)•(4
x
2)
c) (3 x
2)•(4
x
2)
Solution:
a) (3 x)•(4
x) = 12 x
2
b) (3 x)•(4
x
2) = 12 x
3
c) (3 x
2)•(4
x
2) = 12 x
4
EXAMPLE 2.
Monomial times binomial (using
the distributive property!)
a)
(3 x)•(4
x + 3) b) (–3 x
2)•(4
x – 3)
Solution:
a)
(3 x)•(4
x + 3) b) (–3 x
2)•(4
x – 3)
12
x
2 + 9
x –12 x
3 + 9 x
2
EXAMPLE 3.
Monomial times
trinomial
a)
(6 x)•(5
x
2 – 7 x
+ 9) b) (–6 x
2)•(5
x
2 – 7 x
+ 9)
Solution:
a)
(6 x)•(5
x
2 – 7 x
+ 9) b) (–6 x
2)•(5
x
2 – 7 x
+ 9)
30
x
3 – 42 x
2 + 54 x
–30 x
4 + 42 x
3 – 54 x
2
EXERCISES:
Multiply the polynomials.
1.
(2 x)
•(7
x)
2. (5
x)•(9
x2
3. (3 x2)•(8
x3)
4. (3
x)•(4
x + 9)
5. (7
x2)•(2
x – 7)
6. (–4
x2)•(9
x2– 5)
122
7.
(2 x)(3
x2 + 9
x – 6)
8. (8
x3)(9
x2 – 6
x – 8)
9.
(–5 x3)(–3
x2 – 12
x + 5)
10. (–9
x4)(6
x2 + 7
x + 9)
Binomial Times a Binomial
One of
the most frequent products in all of mathematics is the product of two
binomials. In order to multiply two binomials, one method is to make
a
substitution of some other variable in place of the first binomial. Then,
as illustrated in the next example, the distributive property can be
applied. The results of this can be summarized in what is known
as the F
O I
L
method, which is explained in the second part of Example 4.
EXAMPLE 4.
Consider the example: (x
+ 2)( x + 3).
Solution:
You could substitute y = (x
+ 2).
Now the example reads y•(
x + 3) = x
•y
+ 3•y
= x (x
+2) + 3(x +2)
=
x2
+ 2 x + 3 x + 6
=
x
2 + 5 x
+ 6.
The
easier way to multiply binomials is known as the
F OI L
method.
F
= First times first
O = Outer times outer
These are usually (but not
always!) like terms,
I =
Inner times inner
in which case, they are
combined.
L = Last times last
EXAMPLE 4.
Multiply
(x
+ 2)( x + 3)
by the
F
OI
L
method.
F
O I
L
Solution:
(x
+ 2)( x + 3) =
x
•
x
+ 3•
x + 2•
x
+
2•3
=
x
2
+ 3
x + 2 x
+
6
=
x
2
+
5
x +
6
123
EXAMPLE 5.
Multiply (x
+ 6)( x –
3)
by the
F
OI
L
method.
F
O I
L
Solution:
(x + 6)( x –
3) =
x
•
x
+ (–3)•
x +
6•
x
+ 6•(–3)
=
x2
–
3 x +
6 x
–
18
= x
2
+
3
x
–
18
EXERCISES:
F
O
I
L
11. (x
+ 5)( x + 2) =
_____
+ _____ + _____
+ _____
=
_____
+
_____
+ _____
12. (x
+ 3)( x + 7) =
_____ + _____ + _____
+ _____
=
_____
+ _____
+ _____
13. (x
+ 5)( x + 9) =
_____
+ _____ + _____
+ _____
=
_____
+
_____
+ _____
14. (x
+ 4)( x + 8) =
_____ + _____ +
_____
+ _____
=
_____
+ _____
+ _____
15. (x
+ 6)( x –
4) = _____________________________
= __________________________
16. (x
– 6)( x +
4) = _____________________________
= __________________________
124
At some
point that this process becomes comfortable to you, you may wish to
combine like terms and do the exercise in one step instead of two steps.
F
O I
L
17. (x
– 6)( x
– 3) =
_____________________________
=
__________________________
18. (x
+ 7)( x –
5) = _____________________________
=
__________________________
19. (x
– 7)( x +
5) = _____________________________
=
__________________________
20. (x
– 9)( x
– 6) =
______________________________
=
___________________________
What
happens when you take the product of two binomials that are the same but
with the opposite sign in the middle? Use these illustrations to
demonstrate.
EXAMPLE 6.
EXAMPLE 7.
EXAMPLE 8.
(x
– 5)( x +
5) (x
– 7)( x +
7 (4 x
–
9)(4 x + 9)
Solutions: x
2 + 5 x
– 5 x
–
25
x
2 + 7 x
– 7 x
– 49 6
x
2 + 36 x
– 36 x
– 81
x
2 –
25 x
2
–
49 16 x
2 –
81
21. (x–3)(x+3)
= __________________ 22. (x–4)(x+
4) = __________________
=
__________________ =
__________________
23. (x
–5)(x+5)
= __________________ 24. (x–6)(x
+ 6) = __________________
25. (x–7)(x+7)
= __________________ 26. (x–8)(x
+ 8) = __________________
27. (x–9)(x+9)
= __________________ 28. (x–10)(x
+10) = ________________
125
29. (3
x – 4)(3 x
+ 4) = 30. (5 x –
3)(5 x + 3) =
31. (5
x – 3y)(5
x + 3y) = 32. (5 x –
12y)(5 x + 12y) =
EXAMPLE 9.
(3 x –
7y)(2 x + 4y)
F
O
I
L
Solution: (3 x –
7y)(2 x + 4y) = 3 x
•2
x + 3 x
•4y
– 2 x
•7y
– 7y•4y
=
6 x
2
+
12
x y
–
14 x y
–
28y2
=
6 x
2
–
2 x y
–
28y2
EXAMPLE 10. (3 x –
7y)(4 x –
9y)
F
O
I
L
Solution: (3 x –
7y)(4 x –
9y) =
3 x
•4
x
– 3 x
•9y
– 4 x
•7y
+ 7y•9y
= 12 x
2
–
27xy –
28xy + 63y2
= 12 x
2
–
55 x y
+ 63y2
EXERCISES:
F
O I
L
33. (3
x + 4)(4 x –
3) = _______________________________
=
___________________________
34. (3
x – 4)(4 x
+ 3) = _______________________________
= ___________________________
126
35. (3
x – 4y)(4
x – 3y)
= 36. (5 x
+ 3y)(4 x + 5y) =
37. (5
x + 3y)(4 x –
5y) = 38. (5
x – 3y)(4
x – 5y) =
How would
you square a binomial? For example, what if you wanted to find (x + 3)2
? Remember, a quantity squared means the quantity times itself. This
means that
(x + 3)2
= (x + 3)(x + 3), which you already learned to do using
FOIL.
EXAMPLE 11.
(x + 3)2
EXAMPLE 12.
(x
–
7)2
Solution:
(x + 3)( x
+ 3) Solution: (x
– 7)( x
– 7)
x
2
+ 3 x
+ 3 x + 9
x
2
– 7 x
– 7 x
+ 49
x
2 + 6 x
+ 9 x
2
– 14 x
+ 49
EXERCISES:
39. (x
+ 5)2
40. (x –
5)2
( )(
) (
)( )
__________________
__________________
41. (x
– 8)2
42.
(x + 8)2
( )(
)
( )( )
__________________
__________________
127
43. (x
+ 9)2
44. (x –
12)2
45.
(x –
13)2
46. (x + 13)2
47. When
you square a binomial, such as (x + 9)2
or (x
– 12)2,
how can you quickly
determine the middle term?
48. When you square a
binomial, how can you tell the sign of the middle term?
EXTRA CHALLENGE:
49. (5 x +
49)2
50. (8 x
– 42)2
Polynomial Times a Polynomial
Consider
now the problem of multiplying a binomial times a trinomial, a trinomial
times a trinomial, or in general, a polynomial times a polynomial. To
multiply a binomial times a trinomial, you must multiply the first
(of the binomial) times each term of the trinomial. Then take the
second (of the binomial) times the trinomial. The next example
illustrates the process.
128
EXAMPLE 13. Multiply
the binomial times the trinomial: (x +
2)( x
2
+ 4 x + 5)
Solution:
Multiply
x times each term
of the trinomial, then multiply
2 times each term.
(x
+ 2)(
x
2 + 4 x
+ 5) = x
•
x
2
+ x
•4
x
+ x
•5
First times
+
2•x2
+ 2•4
x
+ 2•5
Second times
= x
3
+ 4 x
2
+ 5 x First times
+ 2
x2
+ 8 x
+
10
Second times
= x
3 + 6
x
2 + 13
x + 10 Combine like terms
EXAMPLE 14.
Multiply the binomial times the trinomial:
(2 x
– 3)( x
2
– 6 x + 4)
Solution:
Multiply the
2x times each term of the trinomial. Multiply
–3
times each term.
(2
x
– 3)( x
2 –
6 x + 4) = 2 x
3 –
12 x
2 + 8 x
First times
–
3 x2
+ 18 x –
12
Second times
= 2 x
3 –
15 x
2 + 26 x
–12
Combine like terms
EXERCISES:
Multiply the binomial times the
trinomial.
51.
(x + 3)( x
2 + 3
x + 5) =
52.
(x – 5)(
x
2
+ 7 x + 6)
129
53.
(2 x –
5)(3 x
2
– 4 x +
6) 54.
(4 x –
3)(5 x
2
– 6 x
– 8)
Multiplying a trinomial times a trinomial is an easy extension to what you
have just done. In the case of a trinomial times a trinomial, multiply
the first term in the first trinomial times each term in the second
trinomial. Then multiply the second term in the first trinomial
times each term in the second trinomial. Last, multiply the third
term in the first trinomial times each term in the second trinomial.
There should be a total of nine terms. Finally, combine like terms as
before. The next example illustrates this process.
EXAMPLE 15. Multiply
the trinomials: (x2
–
6x
+ 4)(x2
+ 4x + 8).
Solution: (x2
–
6x
+ 4)(x2
+ 4x + 8) =
x4
+ 4x3
+ 8x2
First
–6x3
– 24x2
– 48x
Second
+
4x2
+ 16x + 32
Third
x4
– 2x3
–12x2
– 32x + 32
130
EXERCISES:
Multiply the trinomials.
55.
(x
2
+ 6 x –
4)( x
2 + 4 x
+ 8) 56.
(x
2
+ 3 x –
5)( x
2
– 2 x
– 9)
57.
(x
2 –
5 x – 8)(
x
2 –
x –
6) 58.
(2 x
2 –
3 x – 6)(
x
2 –
5 x – 4)
59.
(3 x
2 –
7 x + 6)(2 x
2 + 6 x
– 4) 60.
(3 x
2 –
7 x – 4)(5
x
2 –
4 x – 6)
131
QUESTION:
How would you find the
cube of a binomial?
Consider:
EXAMPLE
16. (x
+ 2)3
[Note:
Does (x + 2)3
= x3
+ 23
??]
Solution:
Rewrite (x + 2)3
as
(x + 2)(x + 2)(x + 2)
(x + 2)(x2
+ 4x + 4)
x3
+ 4x2
+ 4x
+ 2x2
+ 8x + 8
x3
+ 6x2
+ 12x + 8
No! (x + 2)3
≠
x3
+ 23!
EXERCISES.
Use the method of the previous
example to find the cubes of the binomials.
61. (x
+ 3)3 62. (x
– 3)3
63. (x
– 5)3 64. (x
+ 5)3
132
ANSWERS 2.01
p. 122–132:
1.
14x2;
2. 45x3;
3. 24x5;
4. 12x2+27x;
5. 14x3–49x2;
6. –36x4+20x2;
7. 6x3+18x2–12x; 8. 72x5–48x4–64x3;
9. 15x5+60x4–25x3;
10. –54x6–63x5–81x4;
11. x2+7x+10;
12. x2+10x+21;
13. x2+14x+45;
14. x2+12x+32;
15. x2+2x–24;
16. x2–2x–24;
17. x2–9x+18;
18. x2+2x–35;
19. x2–2x–35;
20. x2–15x+54;
21. x2–9;
22. x2–16;
23. x2–25;
24. x2–36;
25. x2–49;
26. x2–64;
27. x2–81;
28. x2–100;
29. 9x2–16;
30. 25x2–9;
31. 25x2–9y2;
32. 25x2–144y2;
33.
12x2+7x–12;
34. 12x2–7x–12;
35. 12x2–25xy+12y2;
36. 20x2+37xy+15y2;
37.
20x2–13xy–15y2;
38. 20x2–37xy+15y2;
39. x2+10x+25;
40. x2–10x+25;
41. x2–16x+64;
42. x2+16x+64;
43. x2+18x+81;
44. x2–24x+144;
45.x2–26x+169;
46. x2+26x+169;
47.
Twice the product of the first times the
second;
48. “+” gives “+”, “–” gives “–”; 49. 25x2+490x+2401;
50. 64x2–672x+1764;
51. x3+6x2+14x+15;
52. x3+2x2–29x–30; 53. 6x3–23x2+32x–30;
54. 20x3–39x2–14x+24;
55. x4+10x3+28x2+32x–32;
56. x4+x3–20x2
–17x+45;
57. x4–6x3–9x2+38x+48;
58. 2x4–13x3+x2+42x+24;
59. 6x4+4x3–42x2+64x–24;
60. 15x4–47x3–10x2+58x+24;
61. x3+9x2+27x+27;
62. x3–9x2+27x–27;
63. x3–15x2+75x–125;
64. x3+15x2+75x+125.
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