C HAPTER 4 Polynomial and Rational Functions. S ECTION 4.1 Polynomial Functions Determine roots of...
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Transcript of C HAPTER 4 Polynomial and Rational Functions. S ECTION 4.1 Polynomial Functions Determine roots of...
SECTION 4.1
Polynomial FunctionsDetermine roots of polynomial
equationsApply the Fundamental
Theorem of Algebra
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 The greatest exponent of its variable
Leading Coefficient The coefficient with the greatest exponent
1000x18 + 500x10 + 250x5
Degree – 18, Leading Coefficient - 1000
POLYNOMIALS
Polynomial FunctionIf a function is defined by a
polynomial in one variable with real coefficients
F(x) =1000x18 + 500x10 + 250x5
ZerosThe values of x for a polynomial
function where f(x) = 0. Also known as the x-intercepts.
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 – 1State the degree and leading
coefficientDegree 4, leading coefficient of 3
Determine whether -2 is a zero of f(x)No it is not a zero of the polynomial
POLYNOMIALS Polynomial Equation
A polynomial that is set equal to zero Root
The solution for a polynomial equationZero and Root are often used
interchangeably but technically, you find the zero of a function and the root of an equation.
Can be an imaginary number Complex Numbers
Any number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit
Pure Imaginary NumbersThe complex number a + bi when a = 0
and b does not equal 0 and i is the imaginary unit
POLYNOMIALS
Fundamental Theorem of AlgebraEvery polynomial equation with
degree greater than zero has at least one root in the set of complex numbers.
Corollary to the Fundamental Theorem of AlgebraStates that the degree of a
polynomial indicates the number of possible roots of a polynomial equation
POLYNOMIAL GRAPHS
Graphs on pg 207 Positive leading coefficients and degree
greater than 0 (Top Section) Shows maximum number of times the graph of
each type of polynomial may cross the x-axis General shape of a third degree function and
a fourth-degree function. (Bottom Section) The graph of a polynomial function with odd
degree MUST cross the x-axis AT LEAST ONCE The graph of a function with even degree
MAY or MAY NOT cross the x-axis; if it does it will an even number of times
POLYNOMIAL GRAPHS
Each x-intercept represents a real root of the polynomial equation
If a and b are roots of the equation, then using the corollary to the Fundamental Theorem of Algebra, we can find the polynomial equationSet equation up starting with (x-a)(x-
b)=0.
SECTION 4.2
Quadratic Equations Solve quadratic equations Use the discriminate to describe the roots of
quadratic equations
QUADRATIC EQUATIONS
A polynomial equation with a degree of two. Ways to Solve Quadratic Equations:
Graph Factor Completing the square
Completing the square Used to create a perfect square trinomial Useful when the equation can’t be factored (x + b)2= x2 + 2bx + b2
Given first and middle term, find last Square of half the coefficient of the middle term; only
works with the coefficient of the first term is 1
QUADRATIC EQUATIONS
Ex. x2 -6x -16 Graph
Look at x intercepts Factor
(x+2) and (x-8) Set equal to zero, x = -2, 8
Completing the square (x-3) 2 =25 x = -2, 8
Ex. 3x2 +7x + 7 Graph
Look at x intercepts No x intercepts; roots are imaginary numbers
Completing the square (x+ 7/6) 2 =-35/36 x = -7/6+/-i(35) 1/2 /6
QUADRATIC EQUATIONS
Quadratic Formula
Discriminant
-Tells the nature of the roots of a quadratic equation or the zeros of the related quadratic function
QUADRATIC EQUATIONS
b2-4ac >0 Two distinct real roots
b2-4ac=0 Exactly one real root (actually a double root)
b2-4ac<0 No real roots (Two distinct imaginary roots)
QUADRATIC EQUATIONS
Find the discriminant of x2 -4x +15 and describe the nature of the roots of the equation. Then find the roots. Discriminant = -44; D<0 no real roots Roots: 2-i(11) ½ and 2+i(11) ½
Conjugates Suppose a and b are real numbers with b not
equal to 0. If a + bi is a root of a polynomial equation with real coefficients, then a – bi is also a root of the equation. a + bi and a – bi are conjugate pairs
What are other examples? i and –i; -1 + i and -1 – i
QUADRATIC EQUATIONS
Solve 6x2 + x +2 by using graphing, factoring, completing the square, and the quadratic equation. Graphing
The graph does not touch the x-axis no real roots for the equation, can’t determine roots from graph
Factoring No real roots, factoring can’t be solved
Completing the square (x + 1/12) 2 = -47/144 Roots: -1+/-i(47) 1/2/12
Quadratic Equation A = 6, b = 1, c = 2 X= -1+/-i(47) 1/2/12
SECTION 4.3
The Remainder and Factor Theorems Find the factors of polynomials using the
Remainder and Factor Theorem
THE REMAINDER AND FACTOR THEOREMS
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)
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
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
THE REMAINDER AND FACTOR THEOREMS
Determine the binomial factors of x3 – 7x +6 using synthetic division
R 1 0 -7 6
-4
-3
-2
-1
0
1
2
R 1 0 -7 6
-4 1 -4 9 -30
-3 1 -3 -4 0
-2 1 -2 -3 12
-1 1 -1 -6 12
0 1 0 -7 6
1 1 1 -6 0
2 1 2 -3 0
Factors are:X+3, X-1, X-2
THE REMAINDER AND FACTOR THEOREMS
Determine the binomial factors of x3 – 7x +6 using the Factor Theorem Test values F(x) = x3 – 7x +6; Test -1
No because = 12 F(x) = x3 – 7x +6; Test 1
Yes works because = 0, then find depressed polynomial
Depressed polynomial is x2 + x -6 Now Factor depressed polynomial to get other factors Factors to (X-1)&(X-2) All Factors are (X+3),(X-1)&(X-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. Set dividend equal to 0, plug in -3 for X, and then
solve for K K = 8 Check using synthetic division
SECTION 4.4THE RATIONAL ROOT THEOREM
Learn how to identify all possible rational roots of a polynomial equation using the rational root theorem
Determine the number of positive and negative real roots each polynomial function has
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
THE RATIONAL ROOT THEOREM
List the possible roots of 6x3+11x2-3x-2=0 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 RATIONAL ROOT THEOREM
Integral Root Theorem Let xn + a1xn-1 + …+ an-1x + an =0 represent a
polynomial equation that has a leading coefficient of 1, integeral coefficients, and an can’t equal 0. Any rational roots of this equation must be integral factors of an.
Roots have to be a factor of an, the last coefficient
THE RATIONAL ROOT THEOREM Find the roots of x3+8x2+16x+5=0
How many roots are there? 3
What do they have to be factors of according to the integral root theorem? 5 Possible roots: +/-5 and +/-1
Do synthetic division with these roots to check which is a factor. (IE no remainder) Try 5
Doesn’t work, remainder of 410 Try -5
Works, no remainder Factor or use quadratic formula to find the roots of
the depressed polynomial. Roots: -5, -3-(5) 1/2/2, -3+(5)1/2/2
THE RATIONAL ROOT THEOREM
Descartes’ Rule of Signs Suppose P(x) is a polynomial whose terms are
arranged in descending powers of the variable. Then the number of POSITIVE real zeros
of P(x) is the same as the number of changes in sign of the coefficients of the terms or is less than this by an even number.
The number of NEGATIVE real zeros of P(x) is the same as the number of changes in sign of the coefficients of the terms in P(-x) or less than this number by an even number
THE RATIONAL ROOT THEOREM Find the number of possible positive real zeros and the
number of possible negative real zeros for f(x) = 2x5+3x4-6x3+6x2-8x+3 Positive Real Zeros:
4 Changes, 4, 2, or 0 possible positive real zeros Negative Real Zeros:
F(-x) = -2x5+3x4+6x3+6x2+8x+3
One change, 1 possible negative real zero Find Possible zeros
Possible Values of P: +/-1, +/-3
Possible Values of Q: +/-1, +/-2
Possible Values of P/Q: +/-1, +/-3, +/-1/2, +/-3/2
Test using synthetic division or graphing Rational Roots = -3, ½, 1
SECTION 4.5LOCATING ZEROS OF A POLYNOMIAL FUNCTION
Learn to approximate the real zeros of a polynomial function
LOCATING ZEROS OF A POLYNOMIAL FUNCTION
Location Principle Suppose y = f(x) represents a polynomial function with
real coefficients. If a and b are two numbers with f(a) negative and f(b) positive, the functions has at least one zero between a and b.
IE the answer of the equation at that root or the remainder changes signs between two roots.
LOCATING ZEROS OF A POLYNOMIAL FUNCTION
Determine between which consecutive integers the real zeros of F(x) = x3 – 4x2 – 2x + 8 are located. Method 1: Synthetic Division
Test (-3, 5) Method 2: Graphing Calculator
Use Table Function There is a zero at 4, and between -2 and -1, and
between 1 and 2.
LOCATING ZEROS OF A POLYNOMIAL FUNCTION
Approximate the real zeros of f(x) = 12x3-19x2-x+6 to the nearest tenth. How many zeros?
3 How many positive?
2 or 0 How many negative?
-12x3-19x2+x+6 1
Use graphing calculator Table to see where zeros fall Between -1 and 0, between 0 and 1, and between 1
and 2. Use graphing calculator TableSet to change delta from 1
to .1 to better see where 0’s fall Use graph to trace to see 0’s Zeros are at about -.5, .7, 1.4
LOCATING ZEROS OF A POLYNOMIAL FUNCTION
Upper Bound Theorem Suppose c is a positive real number and P(x) is divided
by x – c. If the resulting quotient and remainder have no change in sign, then P(x) has no real zero greater than c. Thus c is an upper bound of the zeros of P(x).
Helps to determine if you have found all real zeros An integer greater than or equal to the greatest real
zero Lower Bound Theorem
If c is an upper bound of the zeros of P(-x), then –c is a lower bound of the zeros of P(x)
An integer less than or equal to the least real zero.
LOCATING ZEROS OF A POLYNOMIAL FUNCTION
Find the upper and lower bound of the zeros of f(x) = x3 + 3x2-5x-10 Find real zeros:
-3.6, -1.4, 2 Interval of upper and lower bound?
-4<=x<=2 Find the upper and lower bound interval for f(x)
= 6x3-7x2-14x+15 -2 <=x<=3
SECTION 4.6RATIONAL EQUATIONS AND PARTIAL FRACTIONS
Learn how to solve rational equations and inequalities.
Learn how to decompose a fraction into partial fractions
SECTION 4.6RATIONAL EQUATIONS AND PARTIAL FRACTIONS
Rational Equation An equation with one or more rational expressions
What is a rational expression? The quotient of two polynomials in the form
f(x)=g(x)/h(x), where h(x) does not equal 0 How do you solve rational equations?
Multiply each side by the
RATIONAL EQUATIONS AND PARTIAL FRACTIONS
Example 1: Solve a2-5 = a2+a+2
a2-1 a+1 What is the LCD?
a2-1 What do we get for a?
a = 3 or -1 Can both of these be our answers?
a can only be 3 because if we plug in -1 to our original equation we get a denominator of 0.
Example 2: Solve X – 2 = 20 . X + 4 x – 1 x2 + 3x - 4 What is the LCD?
x2 + 3x – 4 which factors to (x-1) * (x+4) What is the answer?
7
RATIONAL EQUATIONS AND PARTIAL FRACTIONS
Example 1: Decompose 8y + 7 into partial fractions. y2 + y - 2
First Factor Denominator (y-1) *(y+2)
Then split into two fractions on other side of equals 8y + 7 = A + B y2 + y – 2 (y-1) (y+2)
Multiply each piece by LCD to get rid of fractions 8y + 7 = A(y+2) + B(y-1)
Eliminate B by plugging in 1 for y Solve for A A = 3
Eliminate A by plugging in 2 for y Solve for B B = 3
Re-write fractions by plugging in values found for A and B 8y + 7 = 5 + 3 y2 + y – 2 (y-1) (y+2)
Check to see if the sum of the two fractions equal the original.
RATIONAL EQUATIONS AND PARTIAL FRACTIONS
Example 2: Decompose 6x - 2 into partial fractions x2 -3x – 10
2 + 4 x+2 x-5
RATIONAL EQUATIONS AND PARTIAL FRACTIONS
Rational Inequalities Same as equations but with inequality sign
Example 1: (x-2)(x-1) < 0 (x-3)(x-4)2
For what values is our domain undefined? 3 and 4
What values make this 0? 2 and 1
Plot these points on a number line with dashes at the above values What happens to our number line?
Splits into intervals Test each interval to see if our inequality is true or
false Works for intervals x <1 and 2<x<3
Show Solution on number line
RATIONAL EQUATIONS AND PARTIAL FRACTIONS
Example 2: 2 + 5 > 3 3a 6a 4
Solve for a first, by multiplying by LCD of 12a. A = 2
What is the zero? 2
What is the excluded value? 0
Test intervals Works for intervals 0<a<2
Show Solution on number line
SECTION 4.7RADICAL EQUATIONS AND INEQUALITIES
Learn how to solve equations and inequalities with radicals involved.
RADICAL EQUATIONS AND INEQUALITIES
Radical Equations Equations in which radical expressions include
variables Extraneous Solutions
Solutions that do not satisfy the original equation Check all solutions back into original equation in
order to exclude those that don’t work
RADICAL EQUATIONS AND INEQUALITIES
Example: x = √x+7) +5 Solve for X x = 9 and x = 2 Check that neither are extraneous solutions Only 9 works, Answer: x=9
Example 2: 4 = 3√ x+2)+8 Solve for X x = -66 Check Works, Answer: x=-66
Example: √ x+1) = 1 + √ 2x-12) X = 8
RADICAL EQUATIONS AND INEQUALITIES Radical Inequalities
Same as equations but with inequality signs Example: √ 4x+5) <10
Solve for X X<23.75 Must also find the lower bound to make √ 4x+5) a real
number. Set √ 4x+5) =0 and solve X>-1.25 Solution is -1.25< X<23.75 Check by testing intervals Graph intervals on number line
Example: √ 6x-5) > 4 X > 7/2