Download Unit 3 Polynomials Study Guide

Unit 3 Polynomials
Study Guide
7-5 Polynomials
Part 1: Classifying Polynomials by Terms
Some polynomials have specific names based upon the number of terms they have:
# of Terms
Name
1
Monomial
2
Binomial
3
Trinomial
4 or more
Polynomial
Monomial: any number, variable, or product of numbers and variables
with whole number exponents.
Examples:
5,x,
x
, 3x 4 y 3
4
Binomial: the sum or difference of two unlike monomials
Examples:
4 x 2 + 5 y , − 12 xy 3 − 12 x 3 y , 4 x − 5
Trinomial: the sum or difference of three unlike monomials
Examples:
4 x 2 + 3x 2 y + 5 y , − 2 x 2 + 5 x − 3
Polynomial: the sum or difference of four or more unlike monomials
Examples:
x3 − x 2 + x − 1 , 2g 3 − g 2 + h3 − h 2 + 2
Part 2: Classifying Polynomials by Degree
Some polynomials have specific names based upon the highest degree.
The degree of a monomial is the sum of the exponents on the variables in that monomial.
For example:
6 x 3 y 2 z : The degree of the monomial is 6, because 3 + 2 + 1 = 6
Sum of Exponents
Degree of Polynomial
Example
0
Constant
5, 3, 2
1
Linear
x , 4x , 6y, 3m
2
Quadratic
x 2 , 4 y 2 , mn
3
Cubic
3 x 2 y , 4m 3 , 2mnr
4
Quartic
x 4 , xy 2 z , x 3 y + x 2 y 2
5
Quintic
x 5 , − 3x 2 y 3
6 or more
6th degree, 7th degree, 8th degree…
7 xy 4 z , x 6 + x 3 + 3
Part 3: Writing Polynomials in Standard Form
A polynomial is written in Standard Form when the terms are in order of degree
from greatest to least. If the degrees add up to the same amount, then arrange the
terms alphabetically.
Example:
6x − 7 x5 + 4x 2 + 9
degree: 1
5
2
0
In Standard Form: − 7 x 5 + 4 x 2 + 6 x + 9
***REMEMBER***
ALWAYS TAKE THE SIGN WITH THE TERM!!!
Part 4: Identifying Leading Coefficients of a Polynomial
A leading coefficient is the first coefficient of the polynomial when written in
Standard Form.
Example:
− y 2 + y 6 − 3y 8
The leading coefficient is -3
− 3y8 + y6 − y 2
Exercises:
Find the degree of the polynomial:
1.
62
2.
3 z 6 − 4 z + 12
3.
5k − 5k 2 − 2
6.
8st 3 + 8s 4 t
Classify the polynomial by degree and term:
4.
2s
5.
12v + 6v 4 + 3
Write each polynomial in standard form, then identify the leading coefficient:
7.
2n − 4 − 3n 3
8.
6h − 2 + 2 h 7
9.
2a − a 4 − a 6 + 3a 3
7-6 Adding and Subtracting Polynomials
Part 1: Adding Polynomials
Like Terms: monomials with the same variables raised to the same power.
2x and -3x
4x3y2 and -2x3y2
Examples of like terms:
When adding or subtracting like terms, the only thing that will change is the
COEFFICIENT
Examples:
(3 x 2 − 5 x + 4) + (−2 x 2 − 8)
#1:
Step 1: Line up like terms vertically. Remember to keep the sign with the
term when you move it.
(3 x 2 − 5 x + 4)
+ ( −2 x 2
− 8)
Step 2: Add vertically.
(3 x 2 − 5 x + 4)
+ ( −2 x 2
− 8)
x 2 − 5x − 4
(− 4 xy + 3x − 2 y ) − (3xy + 2 x − 5 y)
#2
Step 1: Distribute the negative (minus) sign onto the second expression and
rewrite.
− 4 xy + 3 x − 2 y − 3 xy − 2 x + 5 y
Step 2: Regroup like terms and add or subtract them.
− 4 xy − 3 xy + 3 x − 2 x − 2 y + 5 y
= −7 xy + x + 3 y
Exercises:
Add or subtract. Write your answers in standard form.
10.
11.
(2 y
3
) (
8 pr 2 + 6 p − 1
+
)
+ 5y2 − 6y + − 5y2 − 4y +1
−7p +3
12.
13.
( − m 3 − m 2 − m − 2) − ( − m 3 − m − 2)
8 pr 2 + 6 p − 1
−
( − 7 p + 3)
7.7 Multiplying Polynomials
To multiply monomials:
multiply the coefficients
multiply the variables with like bases…add the exponents
1.
2.
To multiply a polynomial by a monomial:
distribute the monomial to each term in the polynomial
multiply the coefficients
multiply the variables with like bases…add the exponents
1.
2.
3.
Examples:
#1
(− 5x y )(2 xy )
#2
2 x 2 x 2 + 5 x + 4 2 x 2 x 2 + 2 x(5 x ) + 2 x(4 ) 4 x 3 + 10 x 2 + 8 x
2
(
3
)
(− 5)(2)(x 2 x )(y 3 y ) ( )
− 10 x 3 y 4
To multiply binomials:
Method 1:
Lattice Method
1.
Given: ( x − 3)(2 x + 4 )
Since there are four terms in the problem, draw a box and split it
into quarters. Write one binomial on the top of the box and one on
the side…as shown below.
x
-3
2x
+4
2.
Multiply the terms and fill in the boxes.
x
2x 2x2
-3
-6x
+4 +4x -12
3.
Re-write the polynomial in standard form and simplify by
combining like terms.
2 x 2 − 6 x + 4 x − 12 2 x 2 − 2 x − 12
Method 2:
Vertical Method
Given: (3 x + 2)( x − 5)
1.
Line up the binomials as if you were multiplying two two-digit
numbers.
3x + 2
x−5
2.
Multiply as if you had a ones digit and a tens digit. Simplify by
adding like terms.
3x + 2
x−5
− 15 x − 10
2
3x + 2 x
3 x 2 − 13 x − 10
Method 3:
FOIL
FOIL is an algorithm…mathematical process…that stands for:
F = First…multiply the first terms in each binomial
O = Outside…multiply the outside terms in each binomial
I = Inside…multiply the inside terms in each binomial
L = Last…multiply the last terms in each binomial.
Given: (− 2 x + 1)(− x + 2 )
F:
O:
I:
L:
-2x(-x)= 2x2
-2x(2) = -4x
1(-x) = -x
1(2) = 2
Simplify by writing in standard form and combining like terms.
2x 2 − 5x + 2
Exercises:
Multiply. Write your answers in standard form.
14.
(2 y )(− 5 y 2 − 4 y + 1)
15.
(2 xyz )(− 4 x 2 yz 3 )
16.
− 4x x 3 + 8x 2
(
)
17.
(x − 5)(x 2 − 4)
18.
(2 x − 3)(x 2 − 6 x + 8)
7.8 Special Products of Binomials
***The methods taught in this section are shortcuts for multiplying binomials.***
You can still use any method for multiplying binomials if you like.
Perfect-Square Trinomial: a trinomial that is the product of squaring a
binomial.
Algebraic Definition:
(a + b )2 = (a + b )(a + b ) = a 2 + 2ab + b 2
(a − b )2 = (a − b )(a − b ) = a 2 − 2ab + b 2
Examples:
#1
( x + 4 )2
Step 1: identify a and b
a=x
b=4
Step 2: substitute the values of a and b into the above algebraic
definition
(x )2 + 2( x)(4) + (4) 2
Step 3: simplify
x 2 + 8 x + 16
#2
(2 x − 3)2
Step 1: identify a and b
a = 2x
b = -3
Step 2: substitute the values of a and b into the above algebraic
definition
(2 x )2 + 2(2 x)(−3) + (−3) 2
Step 3: simplify
4 x 2 − 12 x + 9
Difference of Two Squares: a binomial that is the product of multiplying two
binomials with like terms but opposite signs. You’ll notice that both terms in the
product binomial are perfect squares…and you are subtracting them…Hence the
name Difference of Two Squares!!
Algebraic Definition:
(a + b )(a − b)
= a (a − b) + b( a − b)
= a 2 − ab + ab − b 2
= a2 − b2
Examples:
#1
(x + 4 )( x − 4)
Step 1: identify a and b
a=x
b=4
Step 2: substitute the values of a and b into the above algebraic
definition
x( x − 4) + 4( x − 4)
Step 3: simplify
x 2 − 4 x + 4 x − 16
= x 2 − 16
#2
(2 x + 2)(2 x − 2)
Step 1: identify a and b
a = 2x
b=2
Step 2: substitute the values of a and b into the above algebraic
definition
2 x(2 x − 2) + 2(2 x − 2)
Step 3: simplify
4x 2 − 4x + 4x − 4
= 4x 2 − 4
Exercises:
Multiply. Write your answers in standard form.
19.
(z − 3)(z + 3)
20.
(5 x + 6)2
21.
(10 + x )(10 − x )
22.
(7 x − 5)2