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Review 10.1-Review 10.1-10.410.4
PolynomialsPolynomials
• Monomials - a number, a variable, or a product of a number and one or more variables. 4x, 20x2yw3, -3, a2b3, and 3yz are all monomials.
• Polynomials – one or more monomials added or subtracted
• 4x + 6x2, 20xy - 4, and 3a2 - 5a + 4 are all polynomials.
Vocabulary
Like TermsLike Terms
Like Terms refers to monomials that have the same variable(s) but may have different coefficients. The variables in the terms must have the same powers.
Which terms are like? 3a2b, 4ab2, 3ab, -5ab2
4ab2 and -5ab2 are like.
Even though the others have the same variables, the exponents are not the same.
3a2b = 3aab, which is different from 4ab2 = 4abb.
Like TermsLike Terms
Constants are like terms.
Which terms are like? 2x, -3, 5b, 0
-3 and 0 are like.
Which terms are like? 3x, 2x2, 4, x
3x and x are like.
Which terms are like? 2wx, w, 3x, 4xw
2wx and 4xw are like.
A polynomial with only one term is called a monomial. A polynomial with two terms is
called a binomial. A polynomial with three terms is called a trinomial. Identify the
following polynomials:
Classifying Polynomials
Polynomial DegreeClassified by degree
Classified by number of
terms6
–2 x
3x + 1
–x 2 + 2 x – 5
4x 3 – 8x
2 x 4 – 7x
3 – 5x + 1
0
1
1
4
2
3
constant
linear
linear
quartic
quadratic
cubic
monomial
monomial
binomial
polynomial
trinomial
binomial
Add: (x2 + 3x + 1) + (4x2 +5)
Step 1: Underline like terms:
Step 2: Add the coefficients of like terms, do not change the powers of the variables:
Adding PolynomialsAdding Polynomials
(x2 + 3x + 1) + (4x2 +5)
Notice: ‘3x’ doesn’t have a like term.
(x2 + 4x2) + 3x + (1 + 5)
5x2 + 3x + 6
Some people prefer to add polynomials by stacking them. If you choose to do this, be sure to line up the like terms!
Adding PolynomialsAdding Polynomials
(x2 + 3x + 1) + (4x2 +5)
5x2 + 3x + 6
(x2 + 3x + 1)
+ (4x2 +5)
Stack and add these polynomials: (2a2+3ab+4b2) + (7a2+ab+-2b2)
(2a2+3ab+4b2) + (7a2+ab+-2b2)(2a2 + 3ab + 4b2)
+ (7a2 + ab + -2b2)
9a2 + 4ab + 2b2
Adding PolynomialsAdding Polynomials
1) 3x3 −7x( ) + 3x3 +4x( ) = 6x3 −3x
2) 2w2 +w−5( ) + 4w2 +7w+1( )= 6w2 +8w−4
3) 2a3 +3a2 +5a( )+ a3 +4a+3( ) =
3a3 +3a2 +9a+3
• Add the following polynomials; you may stack them if you prefer:
Subtract: (3x2 + 2x + 7) - (x2 + x + 4)
Subtracting PolynomialsSubtracting Polynomials
Step 1: Change subtraction to addition (Keep-Change-Change.).
Step 2: Underline OR line up the like terms and add.
(3x2 + 2x + 7) + (- x2 + - x + - 4)
(3x2 + 2x + 7)
+ (- x2 + - x + - 4)
2x2 + x + 3
Subtracting PolynomialsSubtracting Polynomials
1) x2 −x−4( )− 3x2 −4x+1( )=−2x2 +3x−5
2) 9y2 −3y+1( )− 2y2 +y−9( )= 7y2 −4y+10
3) 2g2 +g−9( )− g3 +3g2 +3( )= −g3 −g2 +g−12
• Subtract the following polynomials by changing to addition (Keep-Change-Change.), then add:
1. Add the following polynomials:(9y - 7x + 15a) + (-3y + 8x - 8a)
Group your like terms.
9y - 3y - 7x + 8x + 15a - 8a
6y + x + 7a
Combine your like terms.
3a2 + 3ab + 4ab - b2 + 6b2
3a2 + 7ab + 5b2
2. Add the following polynomials:(3a2 + 3ab - b2) + (4ab + 6b2)
Add the polynomials.
+X2
11XX
XYYY
YY
1 11
XYY
Y 111
1. x2 + 3x + 7y + xy + 8
2. x2 + 4y + 2x + 3
3. 3x + 7y + 8
4. x2 + 11xy + 8
Line up your like terms. 4x2 - 2xy + 3y2
+ -3x2 - xy + 2y2
_________________________
x2 - 3xy + 5y2
3. Add the following polynomials using column form:
(4x2 - 2xy + 3y2) + (-3x2 - xy + 2y2)
Rewrite subtraction as adding the opposite.
(9y - 7x + 15a) + (+ 3y - 8x + 8a)
Group the like terms.
9y + 3y - 7x - 8x + 15a + 8a
12y - 15x + 23a
4. Subtract the following polynomials:(9y - 7x + 15a) - (-3y + 8x - 8a)
Rewrite subtraction as adding the opposite.
(7a - 10b) + (- 3a - 4b)Group the like terms.
7a - 3a - 10b - 4b4a - 14b
5. Subtract the following polynomials:(7a - 10b) - (3a + 4b)
Line up your like terms and add the opposite.
4x2 - 2xy + 3y2
+ (+ 3x2 + xy - 2y2)--------------------------------------
7x2 - xy + y2
6. Subtract the following polynomials using column form:
(4x2 - 2xy + 3y2) - (-3x2 - xy + 2y2)
Find the sum or difference.(5a – 3b) + (2a + 6b)
1. 3a – 9b
2. 3a + 3b
3. 7a + 3b
4. 7a – 3b
Find the sum or difference.(5a – 3b) – (2a + 6b)
1. 3a – 9b
2. 3a + 3b
3. 7a + 3b
4. 7a – 9b
Find the sum. Write the answer in standard format.
(5x 3 – x + 2 x
2 + 7) + (3x 2 + 7 – 4 x) + (4x
2 – 8 – x 3)
Adding Polynomials
SOLUTI
ON Vertical format: Write each expression in standard form. Align like terms.
5x 3 + 2 x
2 – x + 7
3x 2 – 4 x + 7
– x 3 + 4x
2 – 8+
4x 3 + 9x
2 – 5x + 6
Find the sum. Write the answer in standard format.
(2 x 2 + x – 5) + (x + x
2 + 6)
Adding Polynomials
SOLUTI
ON Horizontal format: Add like terms.
(2 x 2 + x – 5) + (x + x
2 + 6) =(2 x 2 + x
2) + (x + x) + (–5 + 6)
=3x 2 + 2 x + 1
Find the difference.
(–2 x 3 + 5x
2 – x + 8) – (–2 x 2 + 3x – 4)
Subtracting Polynomials
SOLUTION
Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add.
–2 x 3 + 5x
2 – x + 8
–2 x 3 + 3x – 4– Add the opposite
No change –2 x 3 + 5x
2 – x + 8
2 x 3 – 3x + 4+
Find the difference.
(–2 x 3 + 5x
2 – x + 8) – (–2 x 2 + 3x – 4)
Subtracting Polynomials
SOLUTION
Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add.
–2 x 3 + 5x
2 – x + 8
–2 x 3 + 3x – 4–
5x 2 – 4x + 12
–2 x 3 + 5x
2 – x + 8
2 x 3 – 3x + 4+
Find the difference.
(3x 2 – 5x + 3) – (2 x
2 – x – 4)
Subtracting Polynomials
SOLUTION
Use a horizontal format.
(3x 2 – 5x + 3) – (2 x
2 – x – 4)= (3x 2 – 5x + 3) + (–1)(2 x
2 – x – 4)
= x 2 – 4x + 7
= (3x 2 – 5x + 3) – 2 x
2 + x + 4
= (3x 2 – 2 x
2) + (– 5x + x) + (3 + 4)
MultiplyingMultiplyingPolynomialsPolynomials
Distribute and FOIL
Polynomials * Polynomials Polynomials * Polynomials
Multiplying a Polynomial by another Polynomial requires more than one distributing step.
Multiply: (2a + 7b)(3a + 5b)
Distribute 2a(3a + 5b) and distribute 7b(3a + 5b):
6a2 + 10ab 21ab + 35b2
Then add those products, adding like terms:
6a2 + 10ab + 21ab + 35b2 = 6a2 + 31ab + 35b2
Polynomials * Polynomials Polynomials * Polynomials
An alternative is to stack the polynomials and do long multiplication.
(2a + 7b)(3a + 5b)
6a2 + 10ab21ab + 35b2
(2a + 7b)
x (3a + 5b)
Multiply by 5b, then by 3a:(2a + 7b)
x (3a + 5b)When multiplying by 3a, line up the first term under 3a.
+
Add like terms: 6a2 + 31ab + 35b2
Polynomials * Polynomials Polynomials * Polynomials
Multiply the following polynomials:
1) x+5( ) 2x−1( )
2) 3w−2( ) 2w−5( )
3) 2a2 +a−1( ) 2a2 +1( )
Polynomials * Polynomials Polynomials * Polynomials 1) x+5( ) 2x−1( ) (x + 5)
x (2x + -1)
-x + -5
2x2 + 10x+
2x2 + 9x + -5
2) 3w−2( ) 2w−5( )(3w + -2)
x (2w + -5)-15w + 10
6w2 + -4w+
6w2 + -19w + 10
Polynomials * Polynomials Polynomials * Polynomials
3) 2a2 +a−1( ) 2a2 +1( )
(2a2 + a + -1)
x (2a2 + 1)
2a2 + a + -1
4a4 + 2a3 + -2a2+
4a4 + 2a3 + a + -1
Types of PolynomialsTypes of Polynomials
• We have names to classify polynomials based on how many terms they have:
Monomial: a polynomial with one term
Binomial: a polynomial with two terms
Trinomial: a polynomial with three terms
F : Multiply the First term in each binomial. 2x • 4x = 8x2
There is an acronym to help us remember how to multiply two binomials without stacking them.
F.O.I.L.F.O.I.L.
(2x + -3)(4x + 5)
(2x + -3)(4x + 5) = 8x2 + 10x + -12x + -15 = 8x2 + -2x + -15
O : Multiply the Outer terms in the binomials. 2x • 5 = 10x
I : Multiply the Inner terms in the binomials. -3 • 4x = -12x
L : Multiply the Last term in each binomial. -3 • 5 = -15
Use the FOIL method to multiply these binomials:
F.O.I.L.F.O.I.L.
1) (3a + 4)(2a + 1)
2) (x + 4)(x - 5)
3) (x + 5)(x - 5)
4) (c - 3)(2c - 5)
5) (2w + 3)(2w - 3)
Use the FOIL method to multiply these binomials:
F.O.I.L.F.O.I.L.
1) (3a + 4)(2a + 1) = 6a2 + 3a + 8a + 4 = 6a2 + 11a + 4
2) (x + 4)(x - 5) = x2 + -5x + 4x + -20 = x2 + -1x + -20
3) (x + 5)(x - 5) = x2 + -5x + 5x + -25 = x2 + -25
4) (c - 3)(2c - 5) = 2c2 + -5c + -6c + 15 = 2c2 + -11c + 15
5) (2w + 3)(2w - 3) = 4w2 + -6w + 6w + -9 = 4w2 + -9
There are three techniques you can use for multiplying polynomials.
The best part about it is that they are all the same! Huh? Whaddaya mean?
It’s all about how you write it…Here they are!1)Distributive Property2)FOIL3)Box Method
Sit back, relax (but make sure to write this down), and I’ll show ya!
1) Multiply. (2x + 3)(5x + 8)
Using the distributive property, multiply 2x(5x + 8) + 3(5x + 8).
10x2 + 16x + 15x + 24
Combine like terms.
10x2 + 31x + 24
A shortcut of the distributive property is called the FOIL method.
The FOIL method is ONLY used when you multiply 2 binomials. It is an
acronym and tells you which terms to multiply.
2) Use the FOIL method to multiply the following binomials:
(y + 3)(y + 7).
(y + 3)(y + 7). F tells you to multiply the FIRST
terms of each binomial.
y2
(y + 3)(y + 7). O tells you to multiply the OUTER
terms of each binomial.
y2 + 7y
(y + 3)(y + 7). I tells you to multiply the INNER
terms of each binomial.
y2 + 7y + 3y
(y + 3)(y + 7). L tells you to multiply the LAST
terms of each binomial.y2 + 7y + 3y + 21
Combine like terms.
y2 + 10y + 21
Remember, FOIL reminds you to multiply the:
First terms
Outer terms
Inner terms
Last terms
The third method is the Box Method. This method works for every problem!
Here’s how you do it. Multiply (3x – 5)(5x + 2)
Draw a box. Write a polynomial on the top and side of a box. It does not matter which goes where.
This will be modeled in the next problem along with
FOIL.
3x -5
5x
+2
3) Multiply (3x - 5)(5x + 2)
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
15x2 - 19x – 10
3x -5
5x
+2
15x2
+6x
-25x
-10
You have 3 techniques. Pick the one you like the best!
15x2
+6x-25x-10
4) Multiply (7p - 2)(3p - 4)
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
21p2 – 34p + 8
7p -2
3p
-4
21p2
-28p
-6p
+8
21p2
-28p-6p+8
Multiply (y + 4)(y – 3)1. y2 + y – 12
2. y2 – y – 12
3. y2 + 7y – 12
4. y2 – 7y – 12
5. y2 + y + 12
6. y2 – y + 12
7. y2 + 7y + 12
8. y2 – 7y + 12
Multiply (2a – 3b)(2a + 4b)1. 4a2 + 14ab – 12b2
2. 4a2 – 14ab – 12b2
3. 4a2 + 8ab – 6ba – 12b2
4. 4a2 + 2ab – 12b2
5. 4a2 – 2ab – 12b2
5) Multiply (2x - 5)(x2 - 5x + 4)You cannot use FOIL because they are not BOTH binomials. You must use the
distributive property.
2x(x2 - 5x + 4) - 5(x2 - 5x + 4)
2x3 - 10x2 + 8x - 5x2 + 25x - 20
Group and combine like terms.
2x3 - 10x2 - 5x2 + 8x + 25x - 20
2x3 - 15x2 + 33x - 20
x2 -5x +4
2x
-5
5) Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH
binomials. You must use the distributive property or box method.
2x3
-5x2
-10x2
+25x
+8x
-20
Almost done!Go to
the next slide!
x2 -5x +4
2x
-5
5) Multiply (2x - 5)(x2 - 5x + 4) Combine like terms!
2x3
-5x2
-10x2
+25x
+8x
-20
2x3 – 15x2 + 33x - 20
Multiply (2p + 1)(p2 – 3p + 4)1. 2p3 + 2p3 + p + 4
2. y2 – y – 12
3. y2 + 7y – 12
4. y2 – 7y – 12
Example: (x – 6)(2x + 1)
x(2x) + x(1) – (6)2x – 6(1)
2x2 + x – 12x – 6
2x2 – 11x – 6
2x2(3xy + 7x – 2y)
2x2(3xy) + 2x2(7x) + 2x2(–2y)
2x2(3xy + 7x – 2y)
6x3y + 14x2 – 4x2y
(x + 4)(x – 3)
(x + 4)(x – 3)
x(x) + x(–3) + 4(x) + 4(–3)
x2 – 3x + 4x – 12
x2 + x – 12
(2y – 3x)(y – 2)
(2y – 3x)(y – 2)
2y(y) + 2y(–2) + (–3x)(y) + (–3x)(–2)
2y2 – 4y – 3xy + 6x
There are formulas (shortcuts) that work for certain polynomial
multiplication problems.
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
(a - b)(a + b) = a2 - b2
Being able to use these formulas will help you in the future when you have to factor. If you do not remember the formulas, you can always multiply
using distributive, FOIL, or the box method.
Let’s try one!1) Multiply: (x + 4)2
You can multiply this by rewriting this as (x + 4)(x + 4)
ORYou can use the following rule as a shortcut:
(a + b)2 = a2 + 2ab + b2
For comparison, I’ll show you both ways.
1) Multiply (x + 4)(x + 4)
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
x2 +8x + 16
x +4
x
+4
x2
+4x
+4x
+16
Now let’s do it with the shortcut!
x2
+4x+4x+16
Notice you have two of
the same answer?
1) Multiply: (x + 4)2
using (a + b)2 = a2 + 2ab + b2
a is the first term, b is the second term(x + 4)2
a = x and b = 4Plug into the formula
a2 + 2ab + b2
(x)2 + 2(x)(4) + (4)2
Simplify.x2 + 8x+ 16
This is the same answer!
That’s why the 2 is in
the formula!
2) Multiply: (3x + 2y)2
using (a + b)2 = a2 + 2ab + b2
(3x + 2y)2
a = 3x and b = 2y
Plug into the formulaa2 + 2ab + b2
(3x)2 + 2(3x)(2y) + (2y)2Simplify
9x2 + 12xy +4y2
Multiply (2a + 3)2
1. 4a2 – 9
2. 4a2 + 9
3. 4a2 + 36a + 9
4. 4a2 + 12a + 9
Multiply: (x – 5)2
using (a – b)2 = a2 – 2ab + b2
Everything is the same except the signs!
(x)2 – 2(x)(5) + (5)2
x2 – 10x + 25
4) Multiply: (4x – y)2
(4x)2 – 2(4x)(y) + (y)2
16x2 – 8xy + y2
Multiply (x – y)2
1. x2 + 2xy + y2
2. x2 – 2xy + y2
3. x2 + y2
4. x2 – y2
5) Multiply (x – 3)(x + 3)
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
x2 – 9
x -3
x
+3
x2
+3x
-3x
-9
This is called the difference of squares.
x2
+3x-3x-9
Notice the middle terms
eliminate each other!
5) Multiply (x – 3)(x + 3) using (a – b)(a + b) = a2 – b2
You can only use this rule when the binomials are exactly the same except for the sign.
(x – 3)(x + 3)
a = x and b = 3
(x)2 – (3)2
x2 – 9
6) Multiply: (y – 2)(y + 2)(y)2 – (2)2
y2 – 4
7) Multiply: (5a + 6b)(5a – 6b)
(5a)2 – (6b)2
25a2 – 36b2
Multiply (4m – 3n)(4m + 3n)
1. 16m2 – 9n2
2. 16m2 + 9n2
3. 16m2 – 24mn - 9n2
4. 16m2 + 24mn + 9n2
Simplify.1)
2)
2(x 5)
2(m 2)
(x 5)(x 5) 2x 10x 25
(m 2)(m 2) 2m 4m 4
Follow the pattern!2(a b) 2 2 a 2ab b 2(x 5) 2 x 10x 25 2(y 3) 2 y 6y 9
LastTerm
Twice the LastTerm
Square of the Last Term
Difference of Squares.
Multiply.
1)
2)
3)
4)
(x 3)(x 3)
(m 7)(m 7)
(y 10)(y 10)
(t 8)(t 8)
2x 9 2m 49 2y 100
2t 64
Inner and Outer terms cancel!
Multiply.
Example 2: Finding Products in the Form (a – b)2
A. (x – 6)2
(a – b) = a2 – 2ab + b2
(x – 6) = x2 – 2x(6) + (6)2
= x – 12x + 36
Use the rule for (a – b)2.
Identify a and b: a = x and b = 6.
Simplify.
B. (4m – 10)2
Multiply.
Example 2: Finding Products in the Form (a – b)2
C. (2x – 5y )2
D. (7 – r3)2
Check It Out! Example 2
Multiply.
a. (x – 7)2
b. (3b – 2c)2
Check It Out! Example 2c
Multiply.
(a2 – 4)2
(a + b)(a – b) = a2 – b2
A binomial of the form a2 – b2 is called a difference of two squares.
Multiply.
Example 3: Finding Products in the Form (a + b)(a – b)
A. (x + 4)(x – 4)
(a + b)(a – b) = a2 – b2
(x + 4)(x – 4) = x2 – 42
= x2 – 16
Use the rule for (a + b)(a – b).
Identify a and b: a = x and b = 4.
Simplify.
B. (p2 + 8q)(p2 – 8q)
Multiply.
Example 3: Finding Products in the Form (a + b)(a – b)
C. (10 + b)(10 – b)
Check It Out! Example 3
Multiply.a. (x + 8)(x – 8)
b. (3 + 2y2)(3 – 2y2)
Check It Out! Example 3
Multiply.
c. (9 + r)(9 – r)