warm up multiply
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Warm Up Multiply. x 4. 1. x ( x 3 ). 3 x 7. 2. 3 x 2 ( x 5 ). 3. 2(5 x 3 ). 10 x 3. 4. x (6 x 2 ). 6 x 3. 5. xy (7 x 2 ). 7 x 3 y. 6. 3 y 2 (–3 y ). –9 y 3. Objectives. Multiply polynomials. - PowerPoint PPT PresentationTRANSCRIPT
Holt McDougal Algebra 2
3-2 Multiplying Polynomials
Warm UpMultiply.
1. x(x3)
3. 2(5x3)
5. xy(7x2)
6. 3y2(–3y)
7x3y
x4
10x3
–9y3
2. 3x2(x5) 3x7
4. x(6x2) 6x3
Holt McDougal Algebra 2
3-2 Multiplying Polynomials
Multiply polynomials.
Use binomial expansion to expand binomial expressions that are raised to positive integer powers.
Objectives
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
Find each product.
Example 1: Multiplying a Monomial and a Polynomial
A. 4y2(y2 + 3)
Distribute.
B. fg(f4 + 2f3g – 3f2g2 + fg3)
4y2 y2 + 4y2 3
4y2(y2 + 3)
Multiply.4y4 + 12y2
fg(f4 + 2f3g – 3f2g2 + fg3)
Distribute.
Multiply.
fg f4 + fg 2f3g – fg 3f2g2 + fg fg3
f5g + 2f4g2 – 3f3g3 + f2g4
Holt McDougal Algebra 2
3-2 Multiplying Polynomials
Check It Out! Example 1
Find each product.
a. 3cd2(4c2d – 6cd + 14cd2)
Distribute.
b. x2y(6y3 + y2 – 28y + 30)
3cd2 4c2d – 3cd2 6cd + 3cd2 14cd2
3cd2(4c2d – 6cd + 14cd2)
Multiply.12c3d3 – 18c2d3 + 42c2d4
x2y(6y3 + y2 – 28y + 30)
Distribute.
Multiply.
x2y 6y3 + x2y y2 – x2y 28y + x2y 30
6x2y4 + x2y3 – 28x2y2 + 30x2y
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
Find the product.
Example 2A: Multiplying Polynomials
(a – 3)(2 – 5a + a2)
a(a2) + a(–5a) + a(2) – 3(a2) – 3(–5a) –3(2)
Method 1 Multiply horizontally.
a3 – 5a2 + 2a – 3a2 + 15a – 6
a3 – 8a2 + 17a – 6
Write polynomials in standard form.
Distribute a and then –3.
Multiply. Add exponents.
Combine like terms.
(a – 3)(a2 – 5a + 2)
Holt McDougal Algebra 2
3-2 Multiplying Polynomials
(a – 3)(2 – 5a + a2)
Find the product.
Example 2A: Multiplying Polynomials
Method 2 Multiply vertically.
Write each polynomial in standard form.Multiply (a2 – 5a + 2) by –3.
a2 – 5a + 2 a – 3
– 3a2 + 15a – 6
Multiply (a2 – 5a + 2) by a, and align like terms.
a3 – 5a2 + 2a
a3 – 8a2 + 17a – 6 Combine like terms.
Holt McDougal Algebra 2
3-2 Multiplying Polynomials
(y2 – 7y + 5)(y2 – y – 3) Find the product.
Example 2B: Multiplying Polynomials
Multiply each term of one polynomial by each term of the other. Use a table to organize the products.
y4 –y3 –3y2
–7y3 7y2 21y
5y2 –5y –15
y2 –y –3
y2
–7y
5
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)
3b(3b2) + 3b(–2c2) + 3b(–bc) – 2c(3b2) – 2c(–2c2) – 2c(–bc)
Multiply horizontally.
9b3 – 6bc2 – 3b2c – 6b2c + 4c3 + 2bc2
9b3 – 9b2c – 4bc2 + 4c3
Write polynomials in standard form.
Distribute 3b and then –2c.
Multiply. Add exponents.
Combine like terms.
(3b – 2c)(3b2 – 2c2 – bc)
Holt McDougal Algebra 2
3-2 Multiplying Polynomials
(x2 – 4x + 1)(x2 + 5x – 2)
Find the product.Check It Out! Example 2b
Multiply each term of one polynomial by each term of the other. Use a table to organize the products.
x4 –4x3 x2
5x3 –20x2 5x
–2x2 8x –2
x2 –4x 1
x2
5x
–2
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.
x4 + (–4x3 + 5x3) + (–2x2 – 20x2 + x2) + (8x + 5x) – 2
x4 + x3 – 21x2 + 13x – 2
Holt McDougal Algebra 2
3-2 Multiplying Polynomials
4. Find the product. (y – 5)4
Lesson Quiz
2. (2a3 – a + 3)(a2 + 3a – 5) 5jk2 – 10j2k1. 5jk(k – 2j)
2a5 + 6a4 – 11a3 + 14a – 15
y4 – 20y3 + 150y2 – 500y + 625
–0.03x4 – 0.1x3 + 1.27x2 – 0.1x + 10
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.
Find each product.
5. Expand the expression. (3a – b)3 27a3 – 27a2b + 9ab2 – b3