SME
Grade 11 Math · Algebra 2
Polynomial functions: graphs, end behavior, division and the remainder theorem
Use this for “Graph y = (x + 2)²(x − 1)(x − 3): its zeros, their multiplicities and its end behavior.”
Chart
Zeros: x = −2, x = 1, x = 3
y-intercept (0, 12)
f(0) = 12
This is an example. Type your own values to start a new problem.
y
=
a
(x
−
(
)
)
×
(x
−
r₂
)(x
−
r₃
)
x
Input
example
y
Output
worked out (not typed)
Formulas
y
=
a
×
(
x
−
r₁
)
n
×
(
x
−
r₂
)
×
(
x
−
r₃
)
12 = 1 × (0 − (−2))² × (0 − 1) × (0 − 3)
Assumptions
•
y = a(x − r₁)ⁿ(x − r₂)(x − r₃) has degree n + 2: r₁ is a zero n times over.
•
An even multiplicity touches the x-axis and turns; an odd one crosses it, flatter when 3 or more.
•
The leading term decides the ends; a graph of degree d turns at most d − 1 times.
Step-by-step
WE KNOW
a = 1, r₁ = −2, n = 2, r₂ = 1, r₃ = 3, x = 0
FIND
output (y)
Step 1 · Find output (y)
Use y = a × (x − r₁)ⁿ × (x − r₂) × (x − r₃)
Put
x
into
each
factor,
the
repeated
one
n
times,
then
multiply.
y
=
1
×
(0
−
(−2))²
×
(0
−
1)
×
(0
−
3)
y
=
1
×
4
×
(−1)
×
(−3)
y
=
12
Check
Put the numbers back in. Both sides should match.
12 = 1 × (0 − (−2))² × (0 − 1) × (0 − 3) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
Adding, subtracting and multiplying polynomials
Factoring polynomials: GCF, trinomials and special products