Download Lesson 4

Polynomial Functions
Let’s say I asked you to find all the roots of 10x3+9x2–19x+6=0.
How can we solve this?
1. _______
2. _______
3. _______
This function cannot be factored using any method we’ve
previously learned. We do know, though, that if a number is a
zero of the function, it will produce a remainder of zero if
synthetic division is used. But what number do we try? There
are infinitely many numbers out there! Well, here lies the need
for the…
Rational Root Theorem: Let P be a polynomial function with integer
coefficients in standard form. If p/q (in lowest terms) is a root of P(x)
= 0, then p is a factor of the ______________ of P and
q is a factor of the _________________ of P.
p - factors of constant term 6: ___________________
q – factors of leading coefficient 10: _________________
p/q: ______________ Wow! That’s a lot of choices (although smaller
than the previous list of infinitely many numbers). We would have to
choose one, do _____________ and cross our fingers that we have a
remainder of ________. If we do, we are good to go. If we don’t we
have to pick another one and try again.
Anyway, we DO have graphing calculators, so let’s use them!
Ex 1. Let f(x) = 10x3 + 9x2 – 19x + 6 = 0
a) State the degree.
b) Classify by the number of terms
c) State the leading coefficient
d) Find the rational roots
Ex 2. Let f(x) = – 17x2 + 59x + 3x3 – 65 = 0
a) State the degree.
b) Classify by the number of terms
c) State the leading coefficient
d) Find the rational roots
You Try: Let f(x) = – 11x + 8x3 + 10x2 + 2 = 0
a) State the degree.
b) Classify by the number of terms
c) State the leading coefficient
d) Find the rational roots
For real numbers, these things mean the same thing:
“solve the equation”, “find the zeros”,
“find the x-intercepts”, “find the roots”
Ex 4: Let f(x) = x2 + x +3x4 – x3 – 1
a) State the degree.
b) Classify by the number of terms
c) State the leading coefficient
d) Is –2 a zero of the function?
You Try: Let f(x) = x2 - 4x + x3 – 4
a) State the degree.
b) Classify by the number of terms
c) State the leading coefficient
d) Is 2 a root of the function?
Since the x-axis only represents ____ numbers, imaginary roots
cannot be determined by using a ______.
The graph of a polynomial function with ___ degree must cross
the x-axis at least _____ (think end behavior!). The graph of a
function with_____ degree __________ cross the x-axis. If
it does, it will cross an even number of times.
Ex 5: Sketch a
graph with degree
3 that only has
one real root.
Ex 6: Sketch a
graph with degree
3 that only has
three real roots.
You Try: Sketch a
graph with degree
3 that only has
two real roots.
 Every polynomial function of degree n > 1 has exactly n
________ zeros, counting multiplicities.
 If ____ is a root of a polynomial function with real-number
coefficients, then _________ is also.
 You will always have an _____ number of ________ zeros.
Ex 7:
a) Write a polynomial equation of least degree with roots 2, 3i
and –3i.
b) Does the equation have an odd or even degree?
c) How many times does the graph of the related function
cross the x-axis?
You Try:
a) Write a polynomial equation of least degree with roots 4, 3i
and –3i.
b) Does the equation have an odd or even degree?
c) How many times does the graph of the related function
cross the x-axis?
Ex 8: Write a polynomial function, P, with the given
information: P(0) = 120; zeros of p(x) are -3, 2, 4
You Try: Write a polynomial function, P, with the given
information: P(0) = 100; zeros of p(x) are 5, -2, 5
Ex 9: Write a polynomial function, P, with the given
information: P(0) = 120; zeros of p(x) are -3, 2, 4, 5i, -5i
You Try: Write a polynomial function, P, with the given
information: P(0) = 100; zeros of p(x) are -6i, -2, 5
Ex 10: Use the equation 32x3 – 32x2 + 4x – 4 = 0.
a) State the number of complex roots.
b) Factor the polynomial by grouping.
c) Find the roots.
You Try: Use the equation 4x3 – 2x2 + 16x – 8 = 0.
a) State the number of complex roots.
b) Factor the polynomial by grouping.
c) Find the roots.
Ex 11: When a gold ball is hit from a tee with a velocity of
160 ft/s at an angle of 45o with respect to the ground, the
height(in feet) of the ball above the ground is given by h(x) =
x2
, where x is the horizontal distance from the tee.
x
800
a) How far from the tee does the ball strike the ground?
b) Verify your answer using a graph.
You Try: A meteorologist sends a temperature probe on a small
weather rocket through a cloud layer. The launch pad for the
rocket is 2 feet off the ground. The height of the rocket after
launching is modeled by the equation h = -16t2+232t+2, where t
is the elapsed time in seconds.
a) How far will the rocket be after 14 seconds??
b) Verify your answer using a graph.