Transcript Chapter 7

Chapter 7
POLYNOMIAL FUNCTIONS
Polynomial in one variable
 A polynomial in one variable x, is an expression of
the form a0xn + a1xn-1 +….+ an-1x + anx. The
coefficients a0, a1,a2,…, an, represent complex
numbers (real or imaginary), a0 is not zero and n
represents a nonnegative integer.

Example: 1000x18 + 500x10 + 250x5
 Degree
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The greatest exponent of its variable
Also tells the number of zeros or roots
 Leading Coefficient

The coefficient with the greatest exponent
 1000x18 + 500x10 + 250x5

Degree – 18, Leading Coefficient - 1000
Polynomials
 Consider f(x) = x3 + -6x2 + 10x – 8
 State
the degree and leading coefficient.
Degree of 3 and leading coefficient of 1
 Determine whether 4 is a zero of f(x).
Evaluate f(4)
Yes it is a zero.
 Example f(x) = 3x4 – x3 + x2 + x – 1
 State the degree and leading coefficient
Degree 4, leading coefficient of 3
 Determine whether -2 is a zero of f(x)
No it is not a zero of the polynomial
Common polynomial functions
Type
 Constant
 Linear
 Quadratic
 Cubic
 Quartic
 Quintic
Degree
Examples
 0
9
 1
x-2
 2
3x2-3x+4
 3
3x3-6x2-3x+4
 4
x4-2x3-5x2+4x-3
 5
x5+3x4-7x3-x2-x+2
Evaluating Functions
 F(x)=3x2-3x+1
 Find values of F(0)= 3(0)2-3(0)+1=1
 F(1)=
3(1)2-3(1)+1
1
 F(-1)=
 3(-1)2-3(-1)+1
7

Evaluating polynomials
 f(x) = 3x² - 3x + 1
 Evaluate f(a)
 p(x) = x³ +4x² - 5x
 Evaluate p(a²)
 q(x) = x² + 3x =4
 Evaluate q(a + 1)
Graphs of Polynomials
End Behavior
 What happens to f(x), or y, as x approaches infinity
Even degree function
 “ends” in same direction
 May/may not cross x-axis
 If it does, it crosses it an EVEN amount of times
Odd degree function
 “ends” in opposite direction
 Always crosses x-axis at least once
Even Degree
Odd Degree
F(x) = x2
F(x) = x2
F(x) = x3
F(x) =- x3
Determine the end
behavior
Determine whether it is
odd or even
State the number of real
and complex zeros
7.2 Graph using calculator
 Relative minimum – lowest turning point in an
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interval
Absolute Maximum-he least value that a function
assumes over its domain
Relative maximum – highest turning point in an
interval
Absolute Maximum- the greatest value that a
function assumes over its domain
*Find by tracing on calculator
7.3 Solve using quadratic form
 A polynomial can be written in quadratic form if you
square the middle variable and it equals the first
variable
U Substitution
 To write an expression in quadratic form:
 1. Let u = middle variable
 2. Re-write the first variable as u²
 3. Factor
 4. Replace original variable and solve
7.4 Remainder Theorem
 Remainder Theorem
 If a polynomial P(x) is divided by x – r, the remainder is a
constant P(r), and
P(x) =(x-r) * Q(x) + P(r),
where Q(x) is a polynomial with degree one less than the degree
of P(x)
 Use synthetic substitution
 If you divide f(x) by x – a, the remainder is f(a)
The Remainder and Factor Theorems
Quotient
Divisor
Dividend
Remainder
The Remainder and Factor Theorems
 Factor Theorem
 The binomial x – r is a factor of the polynomial P(x) if
and only if P(r) = 0.
 IE. No remainder
 Depressed Polynomial

The quotient when a polynomial is divided by one of its binomial
factors x – r,
 Ex: 2x3 – 3x2 +x divided by x-1
 Is the quotient a factor and/or a depressed
polynomial?
 Yes it is both, 2x2 -x
Factor Theorem
 Do synthetic substitution
 If f(a) has a remainder of 0, then x – a is a factor of
the polynomial
 Factor the depressed polynomial to find the
remaining factors
The Remainder and Factor Theorems
 What is 2x2 + 3x -8 divided by x -2?
 Solve using long division
 Solve using synthetic
 2x + 7 + 6/(x-2)
 Divide x3 – x2 +2 by x +1?
 Solve using long division
 Solve using synthetic
 x2 -2x + 2
Summary of Roots, Zeros, Solutions, Factors etc.
 P(X) Polynomial Function
C
is a ZERO
(x-c) is a factor
 P(x)=0Polunomial Equation
C is a Root or Solution
(x-c) is a factor
The Rational Root Theorem
 Let a0xn + a1xn-1 + …+ an-1x + an =0 represent a
polynomial equation of degree n with integral
coefficients. If a rational number p/q, where
p and q have no common factors, is a root of
the equation, then p is a factor of an and q is a
factor of a0.
 P is a factor of the last coefficient and Q is a
factor of the first coefficient
 P/Q are possible roots of polynomial
7.6 Rational Root Theorem
 To find all possible zeros:
 1. Look at first (Q) and last (P) coefficient
 2. List all (±) factors of last coefficient
 3. List all of these again but divide each last factor
(P) by every last factor (Q)
 4. Ignore repeats
The Rational Root Theorem
 List the possible roots of 6x3+11x2-3x-2=0
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P must be a factor of 2
Q must be a factor of 6
Possible Values of P:
 +/-1, +/-2
Possible Values of Q:
 +/-1, +/-2, +/-3, +/-6
Possible rational roots, p/q :
 +/-1, +/-2, +/-1/2, +/-1/3, +/-1/6, +/-2/3
Use graphing to narrow down the possibilities
 Find zero at X = -2
Check using synthetic, then factor the depressed
polynomial to get roots
 X = -2, -1/3, 1/2
The Remainder and Factor Theorems
 Determine the binomial factors of x3 -2x2-13x-
10

X+1, X+2, X-5
 Find the value of K so that the remainder of
(x3 + 3x2 – kx – 24) divided by (x + 3) is 0.
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Set dividend equal to 0, plug in -3 for X, and then
solve for K
K=8
Check using synthetic division
To find all zeros…
 1. Find one the hard way
 2. Use synthetic substitution to find a depressed
polynomial
 3. Factor answer and set each equal to zero to find
the other zeros
Hints for finding zeros
 Try to trace all on calculator first
 Find one on calc, use this for synthetic substitution
 If you know that one zero is a complex #, another
zero is ALWAYS its conjugate (find the other by
graphing or rational zero theorem)
Operations with Functions
Sum: (f+g)(x)=f(x) + g(x)
Difference: (f-g)(x)=f(x) - g(x)
Product: (f*g)(x)=f(x) * g(x)
Quotient: f  =f(x) / g(x)
  x 
g
Composition of Functions
 (f o g)(x) means f[g(x)]
 f[g(x)] means to substitute the function g(x)
wherever you see an x in f(x)
Inverse relations
 To find the inverse of a relation, flip-flop x and y in
each ordered pair
 To find the inverse, f ˉ¹(x), of a function:
 1. Replace f(x) with y
 2. Interchange x and y
 3. Solve for y
 4. Replace y with f ˉ¹(x)
Inverse Functions
 Ex. F(x) = (x + 3)2 - 5
 F(x)-1
= -3 + -(x+5) 1/2
 Ex. F(x) = 1/(x)3
 F(x)-1 =1/(x)1/3
 The graphs of inverse functions are symmetric across
the line y = x
To determine if two functions are inverses…
Method 1
 Inverses if (f o g)(x) = x and (g o f)(x) = x
Method 2
 Each inverse is the other function