Algebra 1: Chapter 13 Notes
Lesson 13.1
Introduction to Polynomials
Vocabulary
Polynomial
Each piece of a polynomid being added or subtracted
Terms
Coefficient
The number being multiplied by the variable in each term (leading is infront of term w/ highest step)
Constant
Any number not attached to a variable and is its own item
Standard
Form
Term w/ the largest exponent comes first then terms go in descending order.
Highest exponent in a polynomial
Degree
Monomial
A polynomial w/ 1 term
Binomial
A polynomial w/ 2 terms
Trinomial
A polynomial w/ 3 terms
Examples:
a)
An expression w/ variables, coefficients, & exponents. Exponents must be positive whole numbers
Determine whether each expression is a polynomial. Use the definition of a polynomial in your
notes to help you. If it is a polynomial, name the polynomial, identify the terms, coefficients, and
degree of the polynomial.
2
2
3
𝑥−1
𝑥−2
b)
yes
Polynomial? _________
yes
Polynomial? _________
No
Polynomial? _________
# of Terms: _______
# of Terms: _______
# of Terms: _______
Name: ______________
Name: ______________
Name: ______________
Coefficients: ________
Coefficients: ________
Coefficients: ________
Constants: __________
Constants: __________
Constants: __________
Degree: _______
Degree: _______
Degree: _______
Standard Form: __________________
Standard Form: __________________
Standard Form: __________________
Leading Coefficient: _______
Leading Coefficient: _______
Leading Coefficient: _______
− 2𝑚 + 4𝑚 − 8
c)
𝑥
− 2𝑥 + 100𝑥
5 +4
+3
1
Lesson 13.2
Adding & Subtracting Polynomials
Warm Up: Recall from previous chapters, simplify the following (“combine like terms”):
#1)
2
2
#2)
𝑥 + 2𝑦 + 4𝑥 − 3𝑦
2
2
2
𝑥 − 𝑥 + 4𝑥 + 8𝑥 − 6𝑥
Step #1:
Substitute correct functions -- USE PARENTHESIS
Step #2:
Distribute the (-) if necessary & drop the parenthesis -- only for subtracting
Step #3:
Combine like terms -- add / subtract coefficients, keep the variable and exponent the same
Step #4:
Write in standard form
Examples:
Given the following equations, determine each function. Write your answers in standard form.
3
2
𝐴 (𝑥) = 𝑥 + 5𝑥 − 9
𝐷(𝑥) =
𝐵(𝑥) =
2
𝐶(𝑥) = 2𝑥 + 7𝑥
2
− 2𝑥 − 8𝑥
a)
𝐽(𝑥) = 𝐴(𝑥) + 𝐶(𝑥)
𝐿(𝑥) = 𝐶(𝑥) + 𝐷(𝑥)
d)
2
− 3𝑥 − 𝑥 + 1
M(x) = B(x) - A(x)
b)
e)
𝐾(𝑥) = 𝐷(𝑥) − 𝐵(𝑥)
c)
N(x) = A(x) - C(x) - D(x)
2
Lesson 13.3
Multiplying Polynomials
Warm Up: Recall from previous chapters, simplify the following:
3
4
3
2
5
#1)
(𝑥 )(𝑥 )
#2)
(2𝑥 )(4𝑥 )
#3)
(− 3𝑥)(2𝑥 )
#4)
− 3(𝑥 + 4)
#5)
𝑥(2𝑥 + 1)
#6)
− 2𝑥(3𝑥 + 7)
Method 1: Distributive Property
Step #1:
Multiply the first term of the first polynomial by each term in the second polynomial
Step #2:
Multiply the second term of the first polynomial by each term in the second polynomial
Step #3:
Add these two expressions and combine like terms, write in standard form
Examples:
Use the distributive property to multiply the following polynomials.
a)
(5𝑥 − 1) (2𝑥 + 1)
c)
(𝑥 + 2)(𝑥 − 9)
b)
(𝑥 − 7)(𝑥 + 7)
d)
2
2
(2𝑥 + 1)(3𝑥 + 𝑥 − 1)
3
Method 2: Box Method
Step #1:
Write the number of terms in each polynomial
Step #2:
Draw a box with dimensions of step #1
Step #3:
Write polynomials on the outside of the box with a term for each box
Step #4:
Multiply in a grid pattern
Step #5:
Combine like terms & write in standard form
Examples:
Use the box method to multiply the following polynomials
a)
(3𝑥 + 2) (𝑥 − 4)
b)
(𝑥 − 5)(𝑥 + 5)
c)
(2𝑥 + 3)
2
d)
(4𝑥 + 𝑥 − 1)(3𝑥 − 7)
e)
(𝑥 + 2𝑥 − 6)(2𝑥 − 3𝑥 + 5)
2
2
2
4
Lesson 13.4
Simplifying Radicals
Prime Factorization – factoring a number into all _____________ numbers.
Prime Number – a number only divisible by _______ and _________.
Steps to simplifying radicals:
#1)
Complete the __________ _______________________ of the number under the radical.
#2)
Circle all the ___________ of the same number – place these on the ______________ of the radical.
**Write 1 number for each pair of numbers**
#3)
Write all the leftover numbers _____________ the radical & _____________ them.
Examples: Simplify the radicals:
#1)
120
#2)
20
#3)
18
#4)
116
− 8 108
#6)
4 54
3
#8)
24𝑛
#5)
#7)
8𝑝
2
5
#9)
3 2
− 2 72𝑎 𝑏
#10)
2
6 125𝑎 𝑏
6