3-2
3-2 Multiplying Polynomials
Warm Up
Lesson Presentation
Lesson Quiz
Holt
McDougal
Algebra
Holt
McDougal
Algebra
22
3-2
Multiplying Polynomials
Warm Up
Multiply.
1. x(x3) x4
2. 3x2(x5) 3x7
3. 2(5x3) 10x3
4. x(6x2) 6x3
5. xy(7x2)
7x3y
6. 3y2(–3y)
–9y3
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Objectives
Multiply polynomials.
Use binomial expansion to expand
binomial expressions that are raised to
positive integer powers.
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Essential Question
• How do you use Pascal’s Triangle to
expand binomials? (a + b)n
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
To multiply a polynomial by a monomial, use
the Distributive Property and the Properties
of Exponents.
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Example 1: Multiplying a Monomial and a Polynomial
Find each product.
A. 4y2(y2 + 3)
4y2(y2 + 3)
(4y2  y2 )+ (4y2  3) Distribute.
4y4 + 12y2
Multiply constants and add exponents.
B. fg(f4 + 2f3g – 3f2g2 + fg3)
fg(f4 + 2f3g – 3f2g2 + fg3)
Distribute.
(fg  f4 )+ (fg  2f3g)+(fg  -3f2g2 )+ (fg  fg3)
f5g + 2f4g2 – 3f3g3 + f2g4
Multiply.
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Check It Out! Example 1
Find each product.
a. 3cd2(4c2d – 6cd + 14cd2)
Distribute.
3cd2(4c2d – 6cd + 14cd2)
(3cd2  4c2d)+(3cd2  –6cd) + (3cd2  14cd2)
Multiply.
12c3d3 – 18c2d3 + 42c2d4
b. x2y(6y3 + y2 – 28y + 30)
x2y(6y3 + y2 – 28y + 30)
Distribute.
(x2y  6y3 )+ (x2y  y2)+(x2y  –28y) + (x2y  30)
6x2y4 + x2y3 – 28x2y2 + 30x2y
Multiply.
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
To multiply any two polynomials, use the
Distributive Property and multiply each term in
the second polynomial by each term in the first.
Keep in mind that if one polynomial has m terms
and the other has n terms, then the product has
mn terms before it is simplified.
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Example 2A: Multiplying Polynomials
Find the product.
Method 1
(a – 3)(2 – 5a + a2)
Multiply horizontally.
(a – 3)(a2 – 5a + 2) Write polynomials in standard form.
Distribute a and then –3.
a(a2) + a(–5a) + a(2) – 3(a2) – 3(–5a) –3(2)
a3 – 5a2 + 2a – 3a2 + 15a – 6 Multiply. Add exponents.
a3 – 8a2 + 17a – 6
Holt McDougal Algebra 2
Combine like terms.
3-2
Multiplying Polynomials
Example 2A: Multiplying Polynomials
Find the product.
(a – 3)(2 – 5a + a2)
Method 2
Multiply vertically.
a2 – 5a + 2
a–3
– 3a2 + 15a – 6
a3 – 5a2 + 2a
a3 – 8a2 + 17a – 6
Holt McDougal Algebra 2
Write each polynomial in
standard form.
Multiply (a2 – 5a + 2) by –3.
Multiply (a2 – 5a + 2) by a, and
align like terms.
Combine like terms.
3-2
Multiplying Polynomials
Find the product for multiplying two trinomials.
(y2 – 7y + 5)(y2 – y – 3)
Multiply each term of one polynomial by each term of
the other. Use a table to organize the products.
y2
y2
–y
–3
–y3
–3y2
–7y –7y3
7y2
21y
5
–5y –15
y4
5y2
The top left corner is the first
term in the product. Combine
terms along diagonals to get
the middle terms. The bottom
right corner is the last term in
the product.
y4 + (–7y3 – y3 ) + (5y2 + 7y2 – 3y2) + (–5y + 21y) – 15
y4 – 8y3 + 9y2 + 16y – 15
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Check It Out! Example 2a
Find the product.
(3b – 2c)(3b2 – bc – 2c2)
Multiply horizontally.
(3b – 2c)(3b2 –bc –2c2)
Distribute 3b and then –2c.
3b(3b2) + 3b(–bc) + 3b(–2c2) – 2c(3b2) – 2c(–bc) – 2c(–2c2)
Multiply and Add
exponents.
Combine like terms.
9b3 – 3b2c – 6bc2 – 6b2c + 2bc2 + 4c3
9b3 – 9b2c – 4bc2 + 4c3
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Check It Out! Example 2b
Find the product.
(x2 – 4x + 1)(x2 + 5x – 2)
Multiply each term of one polynomial by each term of
the other. Use a table to organize the products.
x2
–4x
1
The top left corner is the first
2
4
3
2
x
x
–4x
x
term in the product. Combine
terms along diagonals to get
2
5x 5x3 –20x
5x
the middle terms. The bottom
right corner is the last term in
–2 –2x2 8x
–2
the product.
x4 + (–4x3 + 5x3) + (–2x2 – 20x2 + x2) + (8x + 5x) – 2
x4 + x3 – 21x2 + 13x – 2
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Example 3: Business Application
A standard Burly Box is p ft by 3p ft by 4p ft. A
large Burly Box has 1.5 ft added to each
dimension. Write a polynomial V(p) in standard
form that can be used to find the volume of a
large Burly Box.
The volume of a large Burly Box is the product of the
area of the base and height. V(p) = A(p)  h(p)
The area of the base of the large Burly Box is the
product of the length and width of the box.
A(p) = l(p)  w(p)
The length, width, and height of the large Burly Box
are greater than that of the standard Burly Box. l(p)
= p + 1.5, w(p) = 3p + 1.5, h(p) = 4p + 1.5
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Example 3: Business Application
Solve A(p) = l(p)  w(p).
p + 1.5
 3p + 1.5
1.5p + 2.25
3p2 + 4.5p
3p2 + 6p + 2.25
Solve V(p) = A(p)  h(p).
3p2 + 6p + 2.25
 4p + 1.5
4.5p2 + 9p + 3.375
12p3 + 24p2 + 9p
12p3 + 28.5p2 + 18p + 3.375
The volume of a large Burly Box can be modeled by
V(p) = 12p3 + 28.5p2 + 18p + 3.375
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Check It Out! Example 4a
Find the product.
(x + 4)4
(x + 4)(x + 4)(x + 4)(x + 4) Write in expanded form.
(x + 4)(x + 4)(x2 + 8x + 16) Multiply the last two
binomial factors.
(x2 + 8x + 16)(x2 + 8x + 16) Multiply the first two
binomial factors.
Distribute x2 and then 8x and then 16.
x2(x2) + x2(8x) + x2(16) + 8x(x2) + 8x(8x) + 8x(16)
+ 16(x2) + 16(8x) + 16(16)
Multiply.
x4 + 8x3 + 16x2 + 8x3 + 64x2 + 128x + 16x2 + 128x + 256
x4 + 16x3 + 96x2 + 256x + 256
Holt McDougal Algebra 2
Combine like terms.
3-2
Multiplying Polynomials
Expanding binomials by multiplying two
polynomials at a time can often be time
consuming and tedious. A quicker
method for expanding binomials is
through the use of Pascal’s Triangle.
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Notice the coefficients of the variables in the final product of
(a + b)3. These coefficients are the numbers from the third
row of Pascal's triangle. (The top 1 does not count as a row)!
Each row of Pascal’s triangle gives the coefficients of the
corresponding binomial expansion. The pattern in the table
can be extended to apply to the expansion of any binomial
of the form (a + b)n, where n is a whole number.
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
This information is formalized by the Binomial
Theorem, which you will study further in Pre-Calculus.
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Example 5: Using Pascal’s Triangle to Expand
Binomial Expressions
Expand the binomial expression.
Method 1
a. (k – 5)3
1331
Identify the coefficients for n = 3, or row 3.
Start with an exponent of 3 for the first term and descend.
Begin with an exponent of 0 for the second term and
ascend.
[1(k)3(–5)0] + [3(k)2(–5)1] + [3(k)1(–5)2] + [1(k)0(–5)3]
k3 – 15k2 + 75k – 125
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Expand the binomial expression.
Method 2
b. (x + 2)3
Identify the coefficients for n = 3, or row 3.
1331
Create a table using the coefficients as the 1st row.
x3
x3
3
3
1
x2
x
2
22
23
Increase 2nd term
6x2
12x
8
Multiply vertically
Decrease 1st term
(x + 2)3 = x3 + 6x2 + 12x + 8
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Check It Out! Example 5
Expand each expression.
c. (x – 4)5
1 5 10 10 5 1
Identify the coefficients for n = 5, or row 5.
[1(x)5(–4)0] + [5(x)4(–4)1] + [10(x)3(–4)2] + [10(x)2(–4)3]
+ [5(x)1(–4)4] + [1(x)0(–4)5]
x5 – 20x4 + 160x3 – 640x2 + 1280x – 1024
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Check It Out! Example 5
Expand the expression.
d. (3x + 1)4
14641
Identify the coefficients for n = 4, or row 4.
Create a table.
1
4
6
4
(3x)4
(3x)3
(3x)2
(3x)1
11
12
13
14
4(27)x3(1)
6(9)x2(1)
4(3)x(1)
1(1)
1(81)x4
1
81x4 + 108x3 + 54x2 + 12x + 1
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Lesson Quiz
Find each product.
1. 5jk(k – 2j) 5jk2 – 10j2k 2. (2a3 – a + 3)(a2 + 3a – 5)
2a5 + 6a4 – 11a3 + 14a – 15
3. The number of items is modeled by
0.3x2 + 0.1x + 2, and the cost per item is
modeled by g(x) = –0.1x2 – 0.3x + 5. Write a
polynomial c(x) that can be used to model the
total cost. –0.03x4 – 0.1x3 + 1.27x2 – 0.1x + 10
4. Find the product.
(y – 5)4 y4 – 20y3 + 150y2 – 500y + 625
5. Expand the expression.
(3a – b)3 27a3 – 27a2b + 9ab2 – b3
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Essential Question
• How do you use Pascal’s Triangle to expand
binomials? (a + b)n
• Identify the nth row by omitting the top value of 1
as a row or by choosing the row with the value n
as the second entry.
• The exponent of a is n in the first term, and the
exponent decreases by 1 in each successive
term.
• The exponent of b is 0 in the first term, and the
exponent increases by 1 in each successive
term.
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Q: When is a solution not an answer?
A: When you make it in a chemistry lab.
Holt McDougal Algebra 2