Chapter 6 Polynomial Functions and Inequalities. 6.1 Properties of Exponents Negative Exponents For...
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Transcript of Chapter 6 Polynomial Functions and Inequalities. 6.1 Properties of Exponents Negative Exponents For...
6.1 Properties of Exponents
Negative Exponents• For any real number a = 0 and any integer n,
a-n =
– Move the base with the negative exponent to the other part of the fraction to make it positive
na
1
Product of Powers
• am · an =
*Add exponents when you have multiplication with the same base
Quotient of Powers• For any real number a = 0,
= am – n
*subtract exponents when you have division with the same base
n
m
a
a
am+n
Power of a Power• (am)n =
*Multiply exponents when you have a power to a power
Power of a Product(ab)m=
*Distribute the exponents when you have a multiplication problem to a power
amn
ambm
Power of a Quotient
* distribute the exponent to both numerator and denominator, then use other property rules to simplify
Zero Powera0 =
* any number with the exponent zero = 1
m
b
a
n
b
a
1
m
m
b
an
n
a
b
n
n
b
a
6.2 Operations with Polynomials
• Polynomial: A monomial or a sum of monomials
– Remember a monomial is a number, a variable, or the product of a number and one or more variables
Rules for polynomials
• Degree of a polynomial = the highest exponent of all the monomial terms
• Adding and Subtracting = combine like terms
• Multiplying = Distribute or FOIL
6.3 Dividing Polynomials
2
3232
3
12159
xy
xyxyyx 1.
When dividing by a monomial:Divide each term by the denominator separately
2.
ab
baabba
5
10155 4332
Dividing by a polynomial
• Long Division: rewrite it as a long division problem
51522 xxx
1525 2 xxx
1. 2.4642 23 xxxx
Dividing by a polynomial
• Synthetic Division– Step 1: Write the terms of the dividend so that the degrees
of the terms are in descending order. Then write just the coefficients.
– Step 2: Write the constant r of the divisor x – r to the left. Bring the first coefficient down.
– Step 3: Multiply the first coefficient by r . Write the product under the second coefficient. Then add the product and the second coefficient.
– Step 4: Multiply the sum by r. Write the product under the next coefficient and add. Repeat until finished.
– Step 5: rewrite the coefficient answers with appropriate x values
ex 1: Use synthetic division to find (x3 – 4x2 + 6x – 4) ÷ (x – 2).
• Step 1
• Step 2
• Step 3
x3 – 4x2 + 6x – 4 1 - 4 6 - 4
1 – 4 6 – 4
1
1 - 4 6 - 4
1
2
- 2
- 4
2
- 4
- 8
x2 – 2x + 2 – 8
x - 5
6.4 Polynomial Functions
• Polynomial in one variable : A polynomial with only one variable
• Leading coefficient: the coefficient of the term with the highest degree in a polynomial in one variable
• Polynomial Function: A polynomial equation where the y is replaced by f(x)
State the degree and leading cofficient of each polynomial, if it is not a polynomial in
one variable explain why.1. 7x4 + 5x2 + x – 9
2. 8x2 + 3xy – 2y2
3. 7x6 – 4x3 + x-1
4. ½ x2 + 2x3 – x5
Evaluating Functions
• Evaluate f(x) = 3x2 – 3x +1 when x = 3
• Find f(b2) if f(x) = 2x2 + 3x – 1
• Find 2g(c+2) + 3g(2c) if g(x) = x2 - 4
End Behavior
• Describes the behavior of the graph f(x) as x approaches positive infinity or negative infinity.
• Symbol for infinity
To determine if a function is even or odd
• Even functions: arrows go the same direction• Odd functions: arrows go opposite directions
To determine if the leading coefficient is positive or negative
• If the graph goes down to the right the leading coefficient is negative
• If the graph goes up to the right then the leading coeffiecient is positive
The number of zeros • zeros are the same as roots: where the graph crosses
the x-axis– The number of zeros of a function can be equal to the exponent
or can be less than that by a multiple of 2.• Example a quintic function, exponent 5, can have 5, 3 or 1 zeros
• To find the zeros you factor the polynomial
Critical Points• points where the graph changes direction.
– These points give us maximum and minimum values• Relative Max/Min
Put it all together
• For the graph given – Describe the end
behavior– Determine whether it is an
even or an odd degree– Determine if the leading
coefficient is positive or negative
– State the number of zeros
Cont…
• For the graph given – Describe the end behavior– Determine whether it is an
even or an odd degree– Determine if the leading
coefficient is positive or negative
– State the number of zeros
• For the graph given – Describe the end
behavior– Determine whether
it is an even or an odd degree
– Determine if the leading coefficient is positive or negative
– State the number of zeros
• For the graph given – Describe the end
behavior– Determine whether it is
an even or an odd degree
– Determine if the leading coefficient is positive or negative
– State the number of zeros
6.5 Analyze Graphs of Polynomial Functions
• Location Principle: used to find the numbers between which you find the roots/zeros
– Make a table to sketch the graph– Estimate and list the location of all the real
zeros • Zeros are between # and #
• Relative Maximum and Minimum:the y-coordinate values at each turning point in the graph of a polynomial.
*These are the highest and lowest points in the near by area of the graph
At most each polynomial has one less turning point than the degree
Find the location of all possible real zeros. Then name the relative minima and maxima as well as
where they occur
Ex 1. f(x) = x3 – x2 – 4x + 4
Find the location of all possible real zeros. Then name the relative minima and maxima as well as
where they occur
Ex 2. f(x) = x4 – 7x2 + x + 5
6.9 Rational Zero Theorem
Parts of a polynomial function f(x) o Factors of the leading coefficient = qo Factors of the constant = po Possible rational roots =
q
p
Ex 1: List all the possible rational zeros for the given function
a. f(x) = 2x3 – 11x2 + 12x + 9 b. f(x) = x3 - 9x2 – x +105
Finding Zeros of a function
• After you find all the possible rational zeros use guess and check along with synthetic division to find a number that gives you a remainder of 0!
• Then factor and or use the quadratic formula with the remaining polynomial to find any other possible zeros
The volume of a rectangular solid is 1001 in3. The height of the box is x – 3 in. The
width is 4 in more than the height, and the lengthis 6 in more than the height. Find the dimensions
of the solid.