Download 1 Name: Pre-Calculus Notes: Chapter 4

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Pre-Calculus Notes: Chapter 4 - Polynomial and Rational Functions
Section 1 – Polynomial Functions
Polynomial in
One Variable,x
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Degree
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Leading
Coefficient
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Polynomial
Function
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Zeros / Solutions / ________________________________________________________________________
Roots / (x-intercepts)
Example1
Consider the polynomial function f(x) = 3x4 – x3 + x2 + x – 1.
a.
State the degree and leading coefficient of the polynomial.
b.
Determine whether -2 is a zero of f(x).
Imaginary
Number
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Complex
Numbers
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Fundamental
Theroem of
Algebra
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Corollary to the
Fundamental
Theorem of
Algebra
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The general shapes of the graphs of polynomial functions with positive leading coefficients and degree
greater than 0 are shown below:
Since the x-axis only represents real numbers, imaginary roots cannot be determined by using a graph.
What must be true about the number of x-intercepts for a polynomial with an even degree?
What must be true about the number of x-intercepts for a polynomial with an odd degree?
--------------------------------------------------------------------------------------------------------------------------------------------------If you know the roots of a polynomial equation, you can use the corollary to the Fundamental Theorem of
Algebra to find the polynomial equation. That is, if a and b are roots of the equation, the equation must
be (x – a)(x – b) = 0
Example 2
Write a polynomial equation of least degree with roots 2, 3i, and -3i.
Does the equation have an odd or even degree?
How many times does the graph cross the x-axis?
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Example 3
State the number of complex roots of the equation 32x3 – 32x2 + 4x – 4 = 0. Then find the roots and graph
the related function.
Example 4
State the number of complex roots of the equation 9x4 – 35x2 – 4 = 0 and then find the roots.
Example 5
When a golf ball is hit from a tree with a velocity of 160 ft/s at an angle of 45o with respect to the ground
x2
(horizontal), the height (in feet) of the ball above the ground is given by h( x ) = x −
, where x is the
800
horizontal distance from the tee.
a.
How far from the tee does the ball strike the ground?
b.
Verify your answer using a graph.
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Section 2 – Quadratic Equations
There are four methods we can use to solve quadratic equations: graphing, factoring, completing the
square, and the quadratic formula.
Quadratic Formula: x =
− b ± b 2 − 4ac
2a
To determine how many solutions and the type of solutions a quadratic function will yield, we refer to the
discriminant: b2 – 4ac.
Example 1
Solve 3x2 + x – 2 = 0
Example 2
Solve by completing the square.
a.
x2 – 6x – 7 = 0
b.
x2 – 6x + 13 = 0
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Example 3
Find the discriminant of x2 + 2x – 2 = 0 and describe the nature of the roots of the equation. Then solve
the equation by using the Quadratic Formula.
Example 4
A late-night talk show host organized a filming stunt from a 200-ft-tall building in a city’s downtown. She
launched a cantaloupe from the tower’s roof at an upward initial velocity of 70 ft/s. A film crew recorded
the fruit’s messy fall into the roped-off area below. The height of the cantaloupe is given by
h(t) = 70t – 16t2 + 200 where t is the number of seconds since the fruit was launched. How long will it
take the cantaloupe to hit the ground?
Conjugates
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Complex Conjugates
Theorem
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Example 5
Solve x2 – 4x = -15
Example 6
Solve each equation using substitution.
a.
x6 + 7x3 – 8 = 0
b.
y10 – 5y6 + 4y2 = 0
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c.
e2x – ex – 6 = 0
d.
e.
(5x – 4)3 + 7 = 5x + 3
f.
e4x = 13e2x – 36
1
6
+8 =
2
x
x
Section 3 – The Remainder and Factor Theorems
Polynomial Division:
Example 1
Divide f(a) = 2a2 + 3a – 8 by a – 2.
Method 1
Good old-fashioned long division
Method 2
The box
Method 3
Synthetic division
(can only be used if the divisor
can be written in the form x – r)
a − 2 2a 2 + 3a − 8
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Remainder Theorem ______________________________________________________________________________
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Example 2
Divide –y6 + 4y4 + 3y2 + 2y by y + 2 using synthetic division.
Factor Theorem
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Example 3
Use the Remainder Theorem to find the remainder when x3 – x2 – 5x – 3 is divided by x – 3. State
whether the binomial is a factor of the polynomial. Explain.
Example 4
A factor of 2x3 – 3x2 + x is x – 1. Determine the remaining factors.
Example 5
Determine the binomial factors of x3 – 2x2 – 13x – 10.
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Example 6
Find the value of j so that the remainder of x 3 + 6 x 2 − jx − 8 ÷ ( x − 2 ) is 0.
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)
Section 5 – Locating Zeros of a Polynomial Function
Locating Intervals Where the Polynomial is Positive or Negative
The Location
Principle
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Example 1
Determine between which consecutive integers the real zeros of f(x) = 12x3 – 20x2 – x + 6 are located.
Example 2
Approximate the real zeros of f(x) = -3x4 + 16x3 – 18x2 + 5 to the nearest tenth. State the interval(s) on
which f(x) is positive and negative.
Example 3
Approximate the real zeros of f(x) = 12x3 – 19x2 – x + 6 to the nearest tenth. State the interval(s) on
which f(x) is positive and negative.
Example 4
The Toaster Treats Company uses a box with a square bottom to package its product. The height of the
box is 3 inches more than the length of the bottom. Find the dimensions of the box if the volume is 42 in3.
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Section 6 – Rational Equations and Partial Fractions
Rational Equation
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Example 1
Solve for x. Check for extraneous solutions.
a.
5
3
=
x − 2 x −1
b.
1+
8
1
=
x − 16 x − 4
c.
2
20
x
−
= 2
x + 4 x − 1 x + 3x − 4
d.
t +5
3
− 16
+
= 2
t + 1 t − 3 t − 2t − 3
2
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Example 2
Solve
(x − 2)(x − 1) < 0 .
(x − 3)(x − 4)
Example 3
Solve
(x + 2)(x − 3) ≥ 0
(x − 1)(x + 1)2
Example 4
Solve
2
5
3
+
>
3a 6a 4
Interval/
Test Point
Check
Yes/No
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Example 5
Solve
1
1
8
+
>
2b + 1 b + 1 15
Interval/
Test Point
Check
Yes/No
Section 8 – Modeling Real-World Data with Polynomial Functions
Example 1
Determine the type of polynomial function that could be used to represent the data in each scatter plot.
a.
b.
c.
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Example 2
Use a graphing calculator to write a polynomial function to model the set of data.
Example 3
Listed below are the number of fat grams and the corresponding Calories for single servings of several
convenience items.
a. What polynomial function could be used to model these data?
b. Use the model to predict the number of Calories for a similar food
item having 10 grams of fat per serving.
c. Use the model to predict the number of fat grams for a similar food
item having 450 Calories per serving.
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