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Holt McDougal Algebra 1
6-3 Polynomials
Holt McDougal Algebra 1
6-3 Polynomials
A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents.
Holt McDougal Algebra 1
6-3 Polynomials
Monomials NOT a monomial Reason
5+z A sum is not a monomial
2/n
A monomial cannot
have a variable
denominator
4a
A monomial cannot
have a variable
exponent
x-1
The variable must have
a whole number
exponent.
Holt McDougal Algebra 1
6-3 Polynomials
The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0.
Monomial Degree
10 0
3x 1
1+2=3
-1.8m5 5
Holt McDougal Algebra 1
6-3 Polynomials
Example 1: Finding the Degree of a Monomial
Find the degree of each monomial.
A. 4p4q3
The degree is 7.
B. 7ed
The degree is 2.
C. 3
The degree is 0.
Holt McDougal Algebra 1
6-3 Polynomials
Check It Out! Example 1
Find the degree of each monomial.
a. 1.5k2m
The degree is 3.
b. 4x
The degree is 1.
c. 2c3
The degree is 3.
Holt McDougal Algebra 1
6-3 Polynomials
A polynomial is a monomial or a sum or difference of monomials. Each monomial in a polynomial is called a term. The degree of a polynomial is the degree of the term with the greatest degree.
Holt McDougal Algebra 1
6-3 Polynomials
Polynomials
1252 23 xxxLeading
Coefficient
Degree of
polynomial
Constant
term
Holt McDougal Algebra 1
6-3 Polynomials
Special Polynomials
• Binomial
– Polynomial with two terms
• Trinomial
– Polynomial with three terms
Holt McDougal Algebra 1
6-3 Polynomials
Find the degree of each polynomial.
Example 2: Finding the Degree of a Polynomial
A. 11x7 + 3x3
The degree of the polynomial is the greatest degree, 7.
B.
The degree of the polynomial is the greatest degree, 4.
Holt McDougal Algebra 1
6-3 Polynomials
Check It Out! Example 2
Find the degree of each polynomial.
a. 5x – 6
The degree of the polynomial is the greatest degree, 1.
b. x3y2 + x2y3 – x4 + 2
The degree of the polynomial is the greatest degree, 5.
Holt McDougal Algebra 1
6-3 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 McDougal Algebra 1
6-3 Polynomials
Write the polynomial in standard form. Then give the leading coefficient.
Example 3A: Writing Polynomials in Standard Form
6x – 7x5 + 4x2 + 9
–7x5 + 4x2 + 6x + 9. The standard form is The leading
coefficient is –7.
Holt McDougal Algebra 1
6-3 Polynomials
Write the polynomial in standard form. Then give the leading coefficient.
Example 3B: Writing Polynomials in Standard Form
y2 + y6 – 3y
The standard form is The leading coefficient is 1.
y6 + y2 – 3y.
Holt McDougal Algebra 1
6-3 Polynomials
A variable written without a coefficient has a coefficient of 1. Remember “the understood 1”
Remember!
y5 = 1y5
Holt McDougal Algebra 1
6-3 Polynomials
Check It Out! Example 3a
Write the polynomial in standard form. Then give the leading coefficient.
16 – 4x2 + x5 + 9x3
The standard form is The leading
coefficient is 1. x5 + 9x3 – 4x2 + 16.
Holt McDougal Algebra 1
6-3 Polynomials
Check It Out! Example 3b
Write the polynomial in standard form. Then give the leading coefficient.
18y5 – 3y8 + 14y
The standard form is The leading coefficient is –3.
–3y8 + 18y5 + 14y.
Holt McDougal Algebra 1
6-3 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
Name Terms
Monomial
Binomial
Trinomial
Polynomial 4 or more
1
2
3
Holt McDougal Algebra 1
6-3 Polynomials
Classify each polynomial according to its degree and number of terms.
Example 4: Classifying Polynomials
A. 5n3 + 4n cubic binomial.
B. 4y6 – 5y3 + 2y – 9 6th-degree polynomial.
C. –2x linear monomial.
Holt McDougal Algebra 1
6-3 Polynomials
Classify each polynomial according to its degree and number of terms.
Check It Out! Example 4
a. x3 + x2 – x + 2 cubic polynomial.
b. 6 constant monomial.
c. –3y8 + 18y5 + 14y 8th-degree trinomial.
Holt McDougal Algebra 1
6-3 Polynomials Example 2
Tell whether is a polynomial. If it is a polynomial, find
its degree and classify it by the number of its terms.
Otherwise, tell why it is not a polynomial.
Quintic binomial Yes
No; negative exponent
No; variable exponent
Quadratic trinomial Yes
constant monomial Yes
7bc3 + 4b4c
n– 2 – 3
6n4 – 8n
2x2 + x – 5
9
Classify by degree and
number of terms
Is it a polynomial? Expression
a.
b.
c.
d.
e.
Holt McDougal Algebra 1
6-3 Polynomials
A tourist accidentally drops her lip balm off the Golden Gate Bridge. The bridge is 220 feet from the water of the bay. The height of the lip balm is given by the polynomial –16t2 + 220, where t is time in seconds. How far above the water will the lip balm be after 3 seconds?
Example 5: Application Continued
After 3 seconds the lip balm will be 76 feet from the water.
Holt McDougal Algebra 1
6-3 Polynomials
Check It Out! Example 5
What if…? Another firework with a 5-second fuse is launched from the same platform at a speed of 400 feet per second. Its height is given by –16t2 +400t + 6. How high will this firework be when it explodes?
1606 feet
Holt McDougal Algebra 1
6-3 Polynomials Check It Out! Example 4a
Simplify. All variables represent nonnegative numbers.
= xy
Holt McDougal Algebra 1
6-3 Polynomials
Solve for the missing exponent.
16?4 6255 dd
21?56 5000105 aaa