Download 6-1 Polynomials

6-1
Polynomials
A monomial is a number or a product of numbers
and variables with whole number exponents.
A polynomial is a sum or difference of monomials.
Each monomial in a polynomial is a term.
Polynomials have no:
-variables or exponents in denominators,
-roots or absolute values of variables,
-fraction or decimal exponents.
Polynomials: 3x4, 2z12 + 9z3, 1 a7, 0.15x101, 3t2 – t3
2
Not polynomials: 3x, |2b3 – 6b|, 8 2 , 1
, m0.75 – m
5y 2
Holt Algebra 2
6-1
Polynomials
The degree of a monomial is the sum of the
exponents of the variables.
Identify the degree of each monomial.
A. z6
z6
Identify the
exponent.
The degree is 6.
C. 8xy3
8x1y3 Add the
exponents.
The degree is 4.
Holt Algebra 2
B. 5.6
5.6 = 5.6x0 Identify the
exponent.
The degree is 0.
D. a2bc3
a2b1c3
Add the
exponents.
The degree is 6.
6-1
Polynomials
The degree of a polynomial is given by the
term with the greatest degree.
A polynomial with one variable is in standard
form when its terms are written in descending
order by degree.
The leading coefficient is the coefficient of the
first term.
Holt Algebra 2
6-1
Polynomials
A polynomial with two terms is called a
binomial, and a polynomial with three terms is
called a trinomial.
A polynomial can also be classified by its
degree.
Holt Algebra 2
6-1
Polynomials
Example 2: Classifying Polynomials
Rewrite each polynomial in standard form.
Then identify the leading coefficient, degree,
and number of terms. Name the polynomial.
A. 3 – 5x2 + 4x
Write terms in
descending order by
degree.
–5x2 + 4x + 3
Leading coefficient: –5
Degree: 2
B. 3x2 – 4 + 8x4
Write terms in
descending order by
degree.
8x4 + 3x2 – 4
Leading coefficient: 8
Degree: 4
Terms: 3
Name: quadratic trinomial
Holt Algebra 2
Terms: 3
Name: quartic trinomial
6-1
Polynomials
Check It Out! Example 2
Rewrite each polynomial in standard form.
Then identify the leading coefficient, degree,
and number of terms. Name the polynomial.
a. –18x2 + x3 – 5 + 2x
Write terms in
descending order by
degree.
1x3– 18x2 + 2x – 5
Leading coefficient: 1
Degree: 3
Terms: 4
Name: cubic polynomial
with 4 terms
Holt Algebra 2
6-1
Polynomials
A function is increasing when its slope is positive (moving
upward)
A function is decreasing when its slope is negative (moving
downward)
A local maximum occurs where a function changes from
increasing to decreasing (top of a hill)
A local minimum occurs where a function changes from
decreasing to increasing (bottom of a valley)
The absolute maximum is the greatest value of a function.
The absolute minimum is the lowest value of a function.
Holt Algebra 2
6-1
Polynomials
Increasing: (-∞, -1) U (1, ∞)
Decreasing: (-1, 1)
Increasing: (-∞, -2.5) U (0, 2.5)
Decreasing: (-2.5, 0) U (2.5, ∞)
Local maximum: x = -1
Local minimum: x = 1
Local maximum: x = -2.5 and x = 2.5
Local minimum: x = 0
Absolute max: ∞
Absolute min: -∞
Absolute max: f(x) = 4
Absolute min: -∞
Holt Algebra 2
6-1
Polynomials
End behavior of a polynomial is what the graph of
f(x) looks like as x approaches ±∞.
Polynomials with an even degree will have both
end behaviors the same.
If the leading coefficient is positive, then the end
behaviors will be +∞. If negative, -∞.
End behavior:
both -∞
Leading coefficient:
negative
End behavior:
both ∞
Leading coefficient:
positive
Holt Algebra 2
6-1
Polynomials
Polynomials with an odd degree will have opposite end
behaviors.
If the leading coefficient is positive, then the end behavior
at x = -∞ is negative and positive at x = ∞.
The opposite is true if the leading coefficient is negative.
End behavior:
negative at x = - ∞
Leading coefficient:
positive
End behavior:
positive at x = - ∞
Leading coefficient:
negative
Holt Algebra 2