Download polynomials - TangHua2012-2013

Polynomials Lesson 3
Multiplying and Factoring Polynomials using the Distributive Method
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Todays Objectives
 Students will be able to demonstrate an understanding
of the multiplication of polynomial expressions,
including:
 Generalize and explain a strategy for multiplication of
polynomials
 Identify and explain errors in a solution for a
polynomial expression
 Generalize and explain strategies used to factor a
trinomial
 Express a polynomial as a product of its factors
Multiplying Polynomials
 Today we will look at a strategy for multiplying and
factoring polynomials:
 Distributive Method
 When binomials have negative terms, it is difficult to
show their product using algebra tiles, but we can use a
rectangle diagram or the distributive property.
 Algebra tiles are limited to multiplying 2 binomials. The
distributive property can be used to multiply any
polynomials
Distributive Property

Consider the following example: (8 − 𝑏)(3 − 𝑏)

First we multiply each term in the first brackets by both terms in the second
bracket. We call this “distributing”

(8 – 𝑏)(3 – 𝑏) = 8(3 − 𝑏) + (−𝑏)(3 – 𝑏) = 8(3) + 8(−𝑏) +
(−𝑏)(3) + (−𝑏)(−𝑏)

= 24 – 8𝑏 – 3𝑏 + 𝑏2

Next, we combine like terms



Like terms are terms that have the same variables which have the same
degree, or exponent
In this case -8b and -3b are like terms. Both terms have the variable “b”,
with the same degree, or exponent (1). b2 is not a like term because “b” has
an exponent, or degree of 2
= 24 – 11𝑏 + 𝑏2
Example (You do)
 Use the distributive property to find the product of the
binomials 𝑐 + 3 𝑐 + 5
 Solution: 𝑐 2 + 8𝑐 + 15
 Notice from the trinomial:
 8 is the sum of 3 and 5
 15 is the product of 3 and 5
 The leading coefficient of the trinomial is 1, as are the leading
coefficients of the binomial factors
Example
 Factor the trinomial x2 – 4x – 21 into two binomials.
 Answer will be in the form (x + a)(x + b), where we
know the following:
 a + b = -4
 (a)(b) = -21
 So we need to find two numbers that ADD to -4, and
MULTIPLY to -21
Example
 Find all factors of -21
 (1)(-21), (-1)(21), (3)(-7), (-3)(7)
 Find one set of factors that add to -4
 (3)(-7)
 So, a = 3, b = -7  (x + 3)(x – 7)
Example (You do)
 Factor the trinomial z2 – 12z + 35
 Solution: Factors of 35 are (1)(35), (-1)(-35), (5)(7), (-5)(-7)
 Two factors that add to -12 are (-5)(-7)
 So, a = -5, b = -7
 (z – 5)(z – 7)
Example
 List all the factors for the polynomial 2x2 – 10x + 12
 Solution: First, remove the GCF  2(x2 – 5x + 6)
 *you must include the factored out 2 as a factor in your final answer!
 Now, factor the trinomial:
 (x + a)(x + b), a + b = -5, (a)(b) = +6
 Find all the factors of +6  (1)(6), (2)(3), (-1)(-6), (-2)(-3)
 Find one set of factors that add to – 5  (-2)(-3)
 So, a = -2, b = -3  Factors of the polynomial 2x2 – 10x + 12 are:

(2)(x - 2)(x – 3)
Example (You do)
 Factor -4t2 – 16t + 128
 Solution:
 Remove the GCF (-4): −4 𝑡 2 + 4𝑡 − 32
 Find two factors of -32 that add to +4: (-4)(8)
 a = -4, b = 8
 (−4)(𝑡 − 4)(𝑡 + 8)
Distributive Method
 The distributive property can be used to perform any
polynomial multiplication. Each term of one
polynomial must be multiplied by each term of the
other polynomial.
Using the distributive property to multiply
two polynomials
 Multiply (2h + 5)(h2 + 3h – 4)
 Solution: Multiply each term in the trinomial by each term in the
binomial. Write the terms in a list.
 = (2ℎ)(ℎ2 + 3ℎ – 4) + 5(ℎ2 + 3ℎ – 4)
 = (2ℎ)(ℎ2) + (2ℎ)(3ℎ) + 2ℎ(−4) + 5(ℎ2) + 5(3ℎ) + 5(−4)
 = 2ℎ3 + 6ℎ2 − 8ℎ + 5ℎ2 + 15ℎ – 20
 = 2ℎ3 + 6ℎ2 + 5ℎ2 – 8ℎ + 15ℎ – 20
 = 2ℎ3 + 11ℎ2 + 7ℎ – 20
𝑐𝑜𝑚𝑏𝑖𝑛𝑒 𝑡ℎ𝑒 𝑙𝑖𝑘𝑒 𝑡𝑒𝑟𝑚𝑠
Example
(-3f2 + 3f – 2)(4f2 – f – 6)
Use the distributive property. Multiply each term in the 1st
trinomial by each term in the 2nd trinomial. Align like terms.
 -3f2(4f2 – f – 6):

3f(4f2 – f – 6):

-2(4f2 – f – 6):

Add:
-12f4 + 3f3 + 18f2
12f3 – 3f2 – 18f
-8f2 + 2f + 12
-12f4 +15f3 + 7f2 –16f + 12
Check your answers!
 One way to check that your product is correct is to
substitute a number for the variable(s) in both your
answer and the original question. If both sides are equal,
then your answer is correct. For this example, let’s let f = 1.
 = (-3f2 + 3f – 2)(4f2 – f – 6) = -12f4 + 15f3 + 7f2 – 16f + 12
 = [-3(1)2 + 3(1) – 2][4(1)2 – 1 – 6] = -12(1)4 + 15(1)3 + 7(1)2 – 16(1) +
12
 = (-2)(-3) = 6
 =6=6
 Since the left side equals the right side, the product is likely correct.
Example (You do)
 Use the distributive property to multiply (3x – 2y)(4x – 3y + 5)
 Solution:
 = 3x(4x – 3y – 5) – 2y(4x – 3y + 5)
 = 3x(4x) + 3x(-3y) + 3x(-5) – 2y(4x) – 2y(-3y) – 2y(5)
 = 12x2 – 9xy – 15x – 8xy + 6y2 – 10y
 = 12x2 – 9xy – 8xy – 15x + 6y2 – 10y  collect like terms
 = 12x2 – 17xy – 15x + 6y2 – 10y
Example
 Simpify (2c – 3)(c + 5) + 3(c – 3)(-3c + 1)
 Solution:
 Use the order of operations. Multiply before adding and
subtracting. Then, combine like terms.
 = 2c(c + 5) – 3(c + 5) + 3[c(-3c + 1) – 3(-3c + 1)]
 = 2c2 + 10c – 3c – 15 + 3[-3c2 + c + 9c – 3]
 = 2c2 + 7c – 15 + 3[-3c2 + 10c – 3]
 = 2c2 + 7c – 15 – 9c2 + 30c – 9
 = -7c2 + 37c – 24
Example (You do)
 Simplify (3𝑥 + 𝑦 – 1)(2𝑥 – 4) – (3𝑥 + 2𝑦)2
 Solution:
 = −3𝑥2 – 14𝑥 – 10𝑥𝑦 – 4𝑦 + 4 – 4𝑦2
Homework
 pg. 186-187, # 5,7,9,11,15-17, 21