Holt Algebra 1
7-5 Polynomials
Warm Up 4-28-08Evaluate each expression for the given value of x.
1. 2x + 3; x = 2 2. x2 + 4; x = –3
3. –4x – 2; x = –1 4. 7x2 + 2x = 3
Identify the coefficient in each term.
5. 4x3 6. y3
7. 2n7 8. –54
7 13
2 69
4 1
2 –1
Holt Algebra 1
7-5 Polynomials
Classify polynomials and write polynomials in standard form. Evaluate polynomial expressions.
Objectives
Holt Algebra 1
7-5 Polynomials
monomialdegree of a monomialpolynomialdegree of a polynomialstandard form of a polynomialleading coefficient
Vocabulary
binomialtrinomial
quadraticcubic
Holt Algebra 1
7-5 Polynomials
A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents.
The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0.
Holt Algebra 1
7-5 Polynomials
Directions: Find the degree of each monomial.
Holt Algebra 1
7-5 Polynomials
Example 1
A. 4p4q3
The degree is 7. Add the exponents of the variables: 4 + 3 = 7.
B. 7ed
The degree is 2. Add the exponents of the variables: 1+ 1 = 2.C. 3
The degree is 0. Add the exponents of the variables: 0 = 0.
Holt Algebra 1
7-5 Polynomials
The terms of an expression are the parts being added or subtracted. See Lesson 1-7.
Remember!
Holt Algebra 1
7-5 Polynomials
Example 2
a. 1.5k2m
The degree is 3. Add the exponents of the variables: 2 + 1 = 3.
b. 4x
The degree is 1. Add the exponents of the variables: 1 = 1.
b. 2c3
The degree is 3. Add the exponents of the variables: 3 = 3.
Holt Algebra 1
7-5 Polynomials
A polynomial is a monomial or a sum or difference of monomials.
• The degree of a polynomial is the degree of the term with the greatest degree.
Holt Algebra 1
7-5 Polynomials
Directions: Find the degree of each polynomial.
Holt Algebra 1
7-5 Polynomials
Example 3
A. 11x7 + 3x3
11x7: degree 7 3x3: degree 3
The degree of the polynomial is the greatest degree, 7.
Find the degree of each term.
B.
Find the degree of each term.
The degree of the polynomial is the greatest degree, 4.
:degree 3 :degree 4
–5: degree 0
Holt Algebra 1
7-5 Polynomials
Example 4
a. 5x – 6
5x: degree 1Find the degree of
each term.The degree of the polynomial is the greatest degree, 1.
b. x3y2 + x2y3 – x4 + 2
x3y2: degree 5
The degree of the polynomial is the greatest degree, 5.
Find the degree of each term.
–6: degree 0
x2y3: degree 5–x4: degree 4 2: degree 0
Holt Algebra 1
7-5 Polynomials
The terms of a polynomial may be written in any order. However, polynomials that contain only one variable are usually written in standard form.
The standard form of a polynomial that contains one variable is written with the terms in order from greatest degree to least degree. When written in standard form, the coefficient of the first term is called the leading coefficient.
Holt Algebra 1
7-5 Polynomials
Directions:
Step 1: Find the degree of EACH term
Step 2: Write in standard form by arranging terms in descending order
Step 3: Determine the LEADING coefficient.
Write the polynomial in standard form. Then give the leading coefficient.
Holt Algebra 1
7-5 Polynomials
Example 5
6x – 7x5 + 4x2 + 9
Find the degree of each term. Then arrange them in descending order:
6x – 7x5 + 4x2 + 9 –7x5 + 4x2 + 6x + 9
Degree 1 5 2 0 5 2 1 0
–7x5 + 4x2 + 6x + 9.The standard form is The leading coefficient is –7.
Holt Algebra 1
7-5 Polynomials
Example 6
Find the degree of each term. Then arrange them in descending order:
y2 + y6 − 3y
y2 + y6 – 3y y6 + y2 – 3y
Degree 2 6 1 2 16
The standard form is The leading coefficient is 1.
y6 + y2 – 3y.
Holt Algebra 1
7-5 Polynomials
A variable written without a coefficient has a coefficient of 1.
Remember!
y5 = 1y5
Holt Algebra 1
7-5 Polynomials
Example 7
16 – 4x2 + x5 + 9x3
Find the degree of each term. Then arrange them in descending order:
16 – 4x2 + x5 + 9x3 x5 + 9x3 – 4x2 + 16
Degree 0 2 5 3 0235
The standard form is The leading coefficient is 1.
x5 + 9x3 – 4x2 + 16.
Holt Algebra 1
7-5 Polynomials
Example 8
Find the degree of each term. Then arrange them in descending order:
18y5 – 3y8 + 14y
18y5 – 3y8 + 14y –3y8 + 18y5 + 14y
Degree 5 8 1 8 5 1
The standard form is The leading coefficient is –3.
–3y8 + 18y5 + 14y.
Holt Algebra 1
7-5 Polynomials
Some polynomials have special names based on their degree and the number of terms they have.
Degree Name
0
1
2
Constant
Linear
Quadratic
3
4
5
6 or more 6th,7th,degree and so on
Cubic
Quartic
Quintic
NameTerms
Monomial
Binomial
Trinomial
Polynomial4 or more
1
2
3
Holt Algebra 1
7-5 Polynomials
Directions:
Step 1: Determine degree of the polynomial or monomial (term with the greatest degree) – first name
Step 2: Count how many total terms there are – last name
Classify each polynomial according to its degree and number of terms.
Holt Algebra 1
7-5 Polynomials
Example 9
A. 5n3 + 4nDegree 3 Terms 2
5n3 + 4n is a cubic binomial.
B. 4y6 – 5y3 + 2y – 9
Degree 6 Terms 4
4y6 – 5y3 + 2y – 9 is a
6th-degree polynomial.
C. –2xDegree 1 Terms 1
–2x is a linear monomial.
Holt Algebra 1
7-5 Polynomials
Example 10
a. x3 + x2 – x + 2Degree 3 Terms 4
x3 + x2 – x + 2 is a cubic polymial.
b. 6
Degree 0 Terms 1 6 is a constant monomial.
c. –3y8 + 18y5 + 14yDegree 8 Terms 3
–3y8 + 18y5 + 14y is an 8th-degree trinomial.
Holt Algebra 1
7-5 Polynomials
Lesson Summary: Part I
Find the degree of each polynomial.
1. 7a3b2 – 2a4 + 4b – 15
2. 25x2 – 3x4
Write each polynomial in standard form. Then
give the leading coefficient.
3. 24g3 + 10 + 7g5 – g2
4. 14 – x4 + 3x2
4
5
–x4 + 3x2 + 14; –1
7g5 + 24g3 – g2 + 10; 7
Holt Algebra 1
7-5 Polynomials
Lesson Summary: Part II
Classify each polynomial according to its degree and number of terms.
5. 18x2 – 12x + 5 quadratic trinomial
6. 2x4 – 1 quartic binomial
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