Download Introduction to Polynomials

Topic 7:
Polynomials
Table of Contents
1. Introduction to Polynomials
2. Adding & Subtracting Polynomials
3. Multiplying Polynomials
4. Factoring Polynomials
5. Factoring Polynomials, part 2
6. Solving Quadratics with Factoring
7. Factoring by Grouping
8. Completing the Square
Introduction to
Polynomials
Vocab
Monomial: a number, a variable, or a product of
numbers and variables with whole-number
exponents.
Degree of a monomial: is the sum of the
exponents of the variables. A constant has
degree 0.
Example: Degree of a
Monomial
Find the degree of each monomial.
A. 4p4q3
B. 7ed
C. 3
Let’s Practice….
Find the degree of each monomial.
a. 1.5k2m
b. 4x
c. 2c3
Review: Like Terms
You can add or subtract monomials by adding or
subtracting like terms.
Like terms
The variables have the same powers.
4a3b2 + 3a2b3 – 2a3b2
Not like terms
The variables have different powers.
Identify Like Terms
Identify the like terms in each polynomial.
A. 5x3 + y2 + 2 – 6y2 + 4x3
Like terms: ______________________
B. 3a3b2 + 3a2b3 + 2a3b2 – a3b2
Like terms: _______________________
Let’s Practice…
Identify the like terms in each polynomial.
A. 4y4 + y2 + 2 – 8y2 + 2y4
Like terms: ____________________________
B. 7n4r2 + 3n2r3 + 5n4r2 + n4r2
Like terms: ___________________________
Simplify.
Add or Subtract
Monomials
A. 4x2 + 2x2
Add or Subtract
Monomials
Simplify.
B. 3n5m4 - n5m4
Let’s Practice…
Simplify.
A. 2x3 - 5x3
B. 2n5p4 + n5p4
Vocab
Polynomial: an expression of more than two
algebraic terms.
Example: 3x4 + 5x2 – 7x + 1
Degree of a polynomial is the degree of the
term with the greatest power/exponent.
Example: The degree of 3x4 + 5x2 – 7x + 1 is 4.
Degree of a Polynomial
Find the degree of each polynomial.
A. 11x7 + 3x3
B.
Let’s Practice…
Find the degree of each polynomial.
a. 5x – 6
b. x3y2 + x2y3 – x4 + 2
Vocab
Standard form of a polynomial: Polynomial
written with the terms in order from greatest
degree to least degree.
Leading Coefficient: When written in standard
form, the coefficient of the first term is called
the leading coefficient.
Example: 3x4 + 5x2 – 7x + 1 and 3 is the leading
coefficient.
Let’s Practice…
Write the polynomial in standard form. Then give the leading
coefficient.
1. 6x – 7x5 + 4x2 + 9
2. 16 – 4x2 + x5 + 9x3
3. 18y5 – 3y8 + 14y
Special Polynomial Names
By Degree
Degree
Name
0
Constant
1
Linear
2
Quadratic
3
Cubic
4
Quartic
5
6 or more
Quintic
6th,7th,degree and so
on
By # of Terms
Terms
Name
1
Monomial
2
Binomial
3
Trinomial
4 or more
Polynomial
Name the Following…
Classify each polynomial according to its degree and number of
terms.
A. 5n3 + 4n
B. 4y6 – 5y3 + 2y – 9
C. –2x
Let’s Practice…
Classify each polynomial according to its degree and number of
terms.
a. x3 + x2 – x + 2
b. 6
c. –3y8 + 18y5 + 14y
Adding and
Subtracting
Polynomials
Adding and Subtracting
Polynomials
Just as you can perform operations on numbers, you
can perform operations on polynomials. To add or
subtract polynomials, combine like terms.
Remember!
Like terms are constants or terms with the same
variable(s) raised to the same power(s).
Simplifying Polynomials
Combine like terms.
A. 12p3 + 11p2 + 8p3
B. 5x2 – 6 – 3x + 8
Let’s Practice…
Combine like terms.
a. 2s2 + 3s2 + s – 3s2 – 5s
b. 4z4 – 8 + -2z2 +16z4 + 2 + 5z3 – 7
Let’s Practice…
Combine like terms.
c. 2x8 + 7y8 – x8 – 9y7 - 10y7 + y8
d. 9b3c2 + -4b3 + 5c2 + 5b3c2 – 13b3c2
2 Methods: Adding Polynomials
Polynomials can be added in either
vertical or horizontal form.
In vertical form, align
the like terms and add:
5x2 + 4x + 1
+ 2x2 + 5x + 2
7x2 + 9x + 3
In horizontal form, use the
Associative and
Commutative Properties to
regroup and combine like
terms.
(5x2 + 4x + 1) + (2x2 + 5x + 2)
= (5x2 + 2x2 + 1) + (4x + 5x) + (1 + 2)
= 7x2 + 9x + 3
Adding Polynomials
Add.
A. (4m2 + 5) + (m2 – m + 6)
B. (10xy + x) + (–3xy + y)
Let’s Practice…
Add.
Add (5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a).
Subtracting Polynomials
To subtract polynomials, remember that subtracting is
the same as adding the opposite (distributing the
negative).
To find the opposite of a polynomial, you must write
the opposite of each term in the polynomial:
–(2x3 – 3x + 7)= –2x3 + 3x – 7
Subtracting Polynomials
Subtract.
(–10x2 – 3x + 7) – (x2 – 9)
Subtracting Polynomials
Subtract.
(x3 + 4y) – (2x3)
(7m4 – 2m2) – (5m4 – 5m2 + 8)
Let’s Practice…
Subtract.
(9q2 – 3q) – (q2 – 5)
(2x2 – 3x2 + 1) – (x2 + x + 1)
Multiplying
Polynomials
F.O.I.L
Multiplying Polynomials
Each term in the first polynomial, must be
multiplied by each term in the second
polynomial.
Method 1: Distribute
First
Outer
Inner
Last
• Multiply!!!
“F.O.I.L.”
Method 2: Box
Multiply (3x – 5)(5x + 2)
1. Draw a box.
2. Write a polynomial
on the top and side
of a box.
3. Multiply.
4. Combine like terms.
3x
5x
+2
-5
Let’s Practice…
1. (7x – 10)(3x + 8)
2. (2x – 3)(4x - 8)
3. (5x -10)(2x +8)
Multiplying Terms with
Exponents
• When FOILing, add the exponents and
multiply coefficients.
• Add the little numbers and multiply the big
numbers!!!
Example:
(3x2 + 10x)(5x3 – 7x2)
15x5 - 21x4 + 50x4 – 70x3
15x5 + 29x4 – 70x3
Let’s Practice…
1. (7x2 – 10x)(3x3 + 8x2)
2. (2x4 – 3x2)(4x - 8)
3. (5x3 + 2x2)(8x - 7)
Multiplying Larger
Polynomials
Each term in the 1st polynomial must be
multiplied by each term in the 2nd.
Example:
(7x2 + 2x + 8)(4x3 – 9x2)
Method 2:
Multiply: (2x - 5)(x2 - 5x + 4)
Let’s Practice…
1. (5x2 + 7) (2x3 – 5x2 +9)
2. (10x4 – 5x2 + 8) (8x3 -3x -6)
Factoring
Polynomials
Vocab: Factoring
Factoring is rewriting an expression as a
product of factors.
It is the reverse of multiplying
polynomials FOILing.
 x  bx  c
2
To determine the factors, ask yourself…
What two #’s add to
the middle number
AND multiply to the
last number?!?!
Let’s Practice…
x  3x  2
What adds (or subtracts)
to get 3 and multiplies to
get 2?
x 2  7 x  10
What adds (or
subtracts) to get -7 and
multiplies to get 10?
2
x  7 x  44
2
What adds (or subtracts) to
get -7 and multiplies to get
-44?
Let’s Practice…
Factor:
1. x2 + 5x + 6
2. x2 -7x + 10
3. x2 -11x +24
Signs of Factors
x  bx  c
2
b
c
+
+
+,+
-
+,- (The factor w/ the greater absolute
value is -)
+
-
+,- (The factor w/ the greater absolute
value is +)
-
+
-, -
-
Factors
Vocab: GCF
The greatest common factor (GCF) is a common
factor of the terms in the expression.
Example:
9 x 2  9 x  18
4 x 2  8 x  12
Vocab: Prime
If a polynomials is “prime” it means there are no
factors.
Find the prime polynomials below:
1. x2 + 7x + 9
2. x2 + 5x + 4
3. x2 + 9x + 10
Let’s Practice….
Factor.
1. y2 -10y +16
2. r2 -11r +24
3. n2 -15n +56
4. v2 + 5v -36
Let’s Practice…
1. x2 + 12x + 36
2. x2 - 8x + 16
3. -x2 +11x -18
4. 16- x2
Factoring
Polynomials,
Part 2
Expanded Form
Expanded Form
(2 x  1)(3 x  5)  6 x 2  10 x  3x  5
 6 x  13x  5
2
When factoring problems where a ≠ 1, we first
want to get the problem into expanded form
before we try to factor.
Creating Expanded Form
Step 1: Multiply a·c
Step 2: To get to expanded form ask yourself
“What multiplies to get a·c, and
add/subtracts to get to b.”
Example:
1. Expand: 2x2 +9x +7
2. Expand: 3x2 + 2x – 8
Method 1:
Step 3: Write your new factors in place of bx.
Step 4: Group the first two terms together and
the last two terms together.
Step 5: Factor each group
Step 6: Factor again to get the complete
factorization
Method 1:
2
6x
+ 13 x +5
1) Multiply a·c (6·5=30)
2) To get to expanded form ask yourself “What
multiplies to get a·c, and add/subtracts to get to b.”
(10, 3)
3) Write your new factors in place of bx.
(6x2+10x+3x+5)
4) Group the first two terms together and the last two
terms together. [(6x2+10x)+(3x+5)]
5) Factor each group [2x(3x+5)+1(3x+5)]
6) Factor again to get the complete factorization
[(3x+5)(2x+1)]
Method 2:
2
6x
Step 3: Fill in box.
Step 4: Factor
horizontally and
vertically.
Step 5: Terms outside
of box are the solution.
+ 13 x +5
Original 1st Term
Expanded Term 1
Expanded Term 2
Original Last Term
Factor:
2
2x
+ 5x -12
Original 1st Term
Expanded Term 1
Expanded Term 2
Original Last Term
Factor:
2
3x
+ 7x +2
Factor:
2
2x
+ 15x -8
Factor:
2
16x
+ 28x +10
Factoring by
Grouping
Factoring by Grouping
– Using the distributive property to factor polynomials with
four or more terms.
– Terms can be put into groups and then factored---- each
group will have a “like” factor used in regrouping.
Factoring by Grouping
A polynomial can be factored by grouping if all of the
following conditions exist.
1. There are four or more terms.
2. Terms have common factors that can be grouped
together, and
3. There are two common factors that are identical.
Symbols:
ax + bx + ay + by = (ax + bx) + (ay + by)
Group, factor
= x(a + b) + y(a + b)
Regroup
= (x + y)(a + b)
Factor by Grouping
Factor each polynomial by grouping. Check your answer.
4
6h
–
3
4h
+ 12h – 8
Factor by Grouping
Factor each polynomial by grouping.
4
5y
–
3
15y
+
2
y
– 3y
Let’s Practice…
Factor each polynomial by grouping.
3
6b
+
2
8b
+ 9b + 12
Let’s Practice…
Factor each polynomial by grouping.
3
4r
+ 24r +
2
r
+6
Factoring with Opposite Groups
3
2x
–
2
12x
+ 18 – 3x
Let’s Practice…
Factor each polynomial. Check your answer.
2
15x
–
3
10x
+ 8x – 12
Let’s Practice…
Factor each polynomial by grouping.
1. 2x3 + x2 – 6x – 3
2. 7p4 – 2p3 + 63p – 18
Factoring Procedure
Completing the
Square
How to:
1. Rearrange equation so it is in the form ax2 + bx = c
2. Divide every term on both sides by a.
3. Add
𝑏 2
( ) to
2
both sides of the equation.
4. Factor.
5.Square root each side and solve.
Solve:
2
x
+ 6x - 3 = 0
Solve:
2
x
- 12x + 7 = 0
Solve:
2
2x
+ 2x -5 =
2
x
Solve:
2
9x
- 12x - 2 = 0