Embed Size (px)
Citation preview
Holt McDougal Algebra 2
6-2 Multiplying Polynomials6-2 Multiplying Polynomials
Holt McDougal Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
Opener-SAME SHEET-10/19Multiply.
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
6-2 Multiplying Polynomials
Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial.
6-1 Hmwk Quiz
A. 3 – 5x2 + 4x B. 3x2 – 4 + 8x4
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
Multiply polynomials.
Use binomial expansion to expand binomial expressions that are raised to positive integer powers.
Objectives
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
To multiply a polynomial by a monomial, use the Distributive Property and the Properties of Exponents.
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
Find each product.
Example 1: Multiplying a Monomial and a Polynomial
A. 4y2(y2 + 3) B. fg(f4 + 2f3g – 3f2g2 + fg3)
C. 3cd2(4c2d – 6cd + 14cd2)D. x2y(6y3 + y2 – 28y + 30)
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
FOIL1. (3xy + 2)(4x + 2y) 2. (x + y)(x – y)
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
Opener-SAME SHEET-10/20• FOIL
1. (x + 3)(4x2 – 2) 2. (3xy +2)(6x + 4y)
Holt McDougal Algebra 2
6-2 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
6-2 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.
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
(a – 3)(2 – 5a + a2)
Find the product.
Example 2A: Multiplying Polynomials
Method 2 Box Method
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
Find the product.
(3b – 2c)(3b2 – bc – 2c2)
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
Cards• Binomial Side Trinomial Side
Holt McDougal Algebra 2
6-2 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
6-2 Multiplying Polynomials
(x2 – 4x + 1)(x2 + 5x – 2)
Find the product.Check It Out! Example 2b
x4 + x3 – 21x2 + 13x – 2
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
Example 4: Expanding a Power of a Binomial
Find the product.(a + 2b)3
(a + 2b)(a + 2b)(a + 2b)
(a + 2b)(a2 + 4ab + 4b2)
a3 + 6a2b + 12ab2 + 8b3
Holt McDougal Algebra 2
6-2 Multiplying PolynomialsCheck It Out! Example 4a
Find the product.(x + 4)4
(x + 4)(x + 4)(x + 4)(x + 4)
(x + 4)(x + 4)(x2 + 8x + 16)
(x2 + 8x + 16)(x2 + 8x + 16)
Combine like terms.x4 + 16x3 + 96x2 + 256x + 256
Holt McDougal Algebra 2
6-2 Multiplying PolynomialsCheck It Out! Example 4b
Find the product.(2x – 1)3
8x3 – 12x2 + 6x – 1 Combine like terms.
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
Wkst
Holt McDougal Algebra 2
6-2 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.
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
6-2 Multiplying Polynomials
This information is formalized by the Binomial Theorem, which you will study further in Chapter 11.
Holt McDougal Algebra 2
6-2 Multiplying PolynomialsExample 5: Using Pascal’s Triangle to Expand Binomial
ExpressionsExpand each expression.
A. (k – 5)3
B. (6m – 8)3
1 3 3 1
k3 – 15k2 + 75k – 125
216m3 – 864m2 + 1152m – 512
Holt McDougal Algebra 2
6-2 Multiplying Polynomials
Check It Out! Example 5
Expand each expression.
a. (x + 2)3
x3 + 6x2 + 12x + 8
b. (x – 4)5
x5 – 20x4 + 160x3 – 640x2 + 1280x – 1024
Holt McDougal Algebra 2
6-2 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
