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GHSGT Worksheet!!
Polynomial Functions
2.1 (M3)
What is a Polynomial?
• 1 or more terms• Exponents are whole numbers (not a fractional)• Coefficients are all real numbers (no imaginary
#’s)• NO x’s in the denominator or under the radical
• It is in standard form when the exponents are written in descending order.
Classification of a Polynomial
Degree Name Example
-2x5 + 3x4 – x3 + 3x2 – 2x + 6
n = 0
n = 1
n = 2
n = 3
n = 4
n = 5
constant 3
linear 5x + 4
quadratic 2x2 + 3x - 2
cubic 5x3 + 3x2 – x + 9
quartic 3x4 – 2x3 + 8x2 – 6x + 5
quintic
One Term:Monomial
Two Terms:Binomial
Three Terms:Trinomial
3+ Terms:Polynomial
Classify each polynomial by degree and by number of terms.
a) 5x + 2x3 – 2x2
cubic trinomial
b) x5 – 4x3 – x5 + 3x2 + 4x3
quadratic monomial
c) x2 + 4 – 8x – 2x3
d) 3x3 + 2x – x3 – 6x5
cubic polynomial quintic trinomial
e) 2x + 5x7
7th degree binomial
2
3 2) 7f
x x
Not a polynomial
EXAMPLE 1 Identify polynomial functions
4
Decide whether the function is a polynomial function.If so, write it in standard form and state its degree, type, and leading coefficient.
a. h (x) = x4 – x2 + 31
SOLUTION
a. Yes it’s a Polynomial. It is in standard form.
Degree 4 - Quartic Its leading coefficient is 1.
EXAMPLE 1 Identify polynomial functions
Decide whether the function is a polynomial function.If so, write it in standard form and state its degree, type, and leading coefficient.
SOLUTION
237)(. xxxgb
37)( 2 xxxg b. Yes it’s a Polynomial. Standard form is Degree 2 – Quadratic Leading Coefficient is
EXAMPLE 1 Identify polynomial functions
Decide whether the function is a polynomial function.If so, write it in standard form and state its degree, type, and leading coefficient.
c. f (x) = 5x2 + 3x –1 – x
SOLUTION
c. The function is not a polynomial function because the term 3x – 1 has an exponent that is not a whole number.
EXAMPLE 1 Identify polynomial functions
Decide whether the function is a polynomial function.If so, write it in standard form and state its degree, type, and leading coefficient.
d. k (x) = x + 2x – 0.6x5
SOLUTION
d. The function is not a polynomial function because the term 2x does not have a variable base and an exponent that is a whole number.
GUIDED PRACTICE for Examples 1 and 2
Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient.
1. f (x) = 13 – 2x
polynomial function; f (x) = –2x + 13; degree 1, type: linear,leading coefficient: –2
2. p (x) = 9x4 – 5x – 2 + 4
3. h (x) = 6x2 + π – 3x
polynomial function; h(x) = 6x2 – 3x + π ; degree 2, type: quadratic, leading coefficient: 6
not a polynomial function
EXAMPLE 2 Evaluate by direct substitution
Use direct substitution to evaluatef (x) = 2x4 – 5x3 – 4x + 8 when x = 3.
f (x) = 2x4 – 5x3 – 4x + 8
f (3) = 2(3)4 – 5(3)3 – 4(3) + 8
= 162 – 135 – 12 + 8
= 23
Write original function.
Substitute 3 for x.
GUIDED PRACTICE for Examples 1 and 2
Use direct substitution to evaluate the polynomial function for the given value of x.
4. f (x) = x4 + 2x3 + 3x2 – 7; x = –2
5
5. g(x) = x3 – 5x2 + 6x + 1; x = 4
9
Graph Trend based on Degree
• Even degree - end behavior going the same direction
• Odd degree – end behavior (tails) going in opposite directions
• Positive– Odd degree—right side up, left side down– Even degree—both sides up
• Negative– Odd degree—right side down, left side up– Even degree—both sides down
Graph Trend based on
Leading Coefficient
Symmetry: Even/Odd/Neither• First look at degree• Even if it is symmetric respect to y-axis
– When you substitute -1 in for x, none of the signs change
• Odd if it is symmetric with respect to the origin– When you substitute -1 in for x, all of the signs
change.
• Neither if it isn’t symmetric around the y axis or origin
Tell whether it is even/odd/neither
1. f(x)= x2 + 2
2. f(x)= x2 + 4x
3. f(x)= x3
4. f(x)= x3 + x
5. f(x)= x3 + 5x +1
Additional Vocabulary to Review• Domain: set of all possible x values• Range: set of all possible y values• Symmetry: even (across y), odd (around origin),
or neither• Interval of increase (where graph goes up to the
right)• Interval of decrease (where the graph goes
down to the right)• End Behavior: f(x) ____ as x+∞
f(x) ______ as x-∞
Math 3 Book
• Page 67 #1 – 4
• Page 68 #5 – 7
• Page 69 #1 – 5, 8 – 16• a) Classify by degree and # of terms• b) Even, Odd or Neither• c) End Behavior
Blue Algebra 2 Books
• P.429 # 4 –10
– #9 and 10• A) Domain and Range• B) Classify by degree and # of terms• C) Even, Odd or Neither• D) End