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A polynomial is an algebraic expression that is the sum of
terms involving variables that has whole number exponents
In standard form, it is written in descending order of the
degrees.
Standard Form of Polynomial Functions
where 𝑎𝑖 are Reals coefficients and n are whole numbers.
For example, 𝑓 𝑥 = 2𝑥 3 + 4𝑥 2 − 5𝑥 + 7 represents a cubic function
in standard form because the highest exponent is 3 followed by the
quadratic term, linear term, and then the constant.
In polynomials, the constant represents the y-intercept of
the polynomial graph.
Monomial
(1 term)
2
Binomial
( 2 terms)
2𝑥 + 5
Trinomial
(3 terms)
𝑥 2 + 5𝑥 − 4
Polynomials with 4
terms
𝑥 5 + 5𝑥 2 + 3𝑥 + 1
2𝑥 3
−
5
5𝑥 3 − 4𝑥 2
9𝑥 3 − 4𝑥 2 + 5
𝑥3 − 𝑥2 + 𝑥 − 1
• Polynomials may be written in factored form to
explicitly show roots of the polynomials, A.K.A. the xintercepts of the polynomial function.
f(x)=a(x-x1)(x-x2)(x-x3)
where x1, x2, and x3
are the x-intercepts of
the function
A)
B)
C)
Examine the sign and the degree of the leading term to know
how polynomial graph behaves as it moves further left and
further right.
Directions:
1) Find the x-intercepts by putting f(x)= 0
2) Find the y-intercept by putting x=0
3) Choose multiple x-values to find interpolating points
4) Plot all points and connect them with a smooth curve
followed with the expected end-behaviors.
X
Y
-1.5
-11.25
-1
0
-0.5
5.25
0
6
0.5
3.75
1
0
1.5
-3.75
2
-6
2.5
-5.25
3
0
3.5
11.25
4
30
Setting f(x) = 0, we get the x-intercepts 3,-1, and 1
Setting x=0, we get the y-intercept 6.
Try: f(x) = x(x-3) (x+2)
Graphing Standard Form
Directions:
1) Find the y-intercept, which is the constant
2) Factor the polynomial (if possible), then find the x-intercepts
3) Choose multiple x-values to find interpolating points
4) Plot all points and connect them with a smooth curve followed
with the expected end-behaviors.
The y-intercept
𝟒
Graph 𝒚 = −𝟐𝒙 + 𝟓
X
Y
-2
-27
-1.5
-5.125
-1
3
-0.5
4.875
0
5
0.5
4.875
1
3
1.5
-5.125
2
-27
Since the y-values shows symmetry about
x=0 and there is only one term with the
variable in the equation, the graph will
behave similar to a parabola.
X
Y
-1.5
-11.25
-1
0
-0.5
5.25
0
6
0.5
3.75
1
0
1.5
-3.75
2
-6
2.5
-5.25
3
0
3.5
11.25
4
30
Setting f(x) = 0, we get the x-intercepts 3,-1, and 1
Setting x=0, we get the y-intercept 6.
Try: f(x) = x(x-3) (x+2)
Determining the Type of Polynomial
Function by Table Values
Make sure
that X-Values
are evenly
spaced
Take the
difference of
the y-values
Are the
differences
constant?
NO
Add 1 to the degree of
the polynomial.
Continue to take
another difference.
YES
Count the number of time each
difference were made. That sum is
the degree of the polynomial
function.
a)
X
y
b)
-4 -3 -2 -1
-77 -38 -17 -8
X
y
1
1
2
6
3
11
0
-5
4
16
1
-2
5
21
2
7
3
28
6
26
7
31
4
67