Presentation on theme: "5.5 Real Zeros of Polynomial Functions"— Presentation transcript:
15.5 Real Zeros of Polynomial Functions MAT SP 20085.5 Real Zeros of Polynomial FunctionsIn this section, we will study the following topics:The Remainder and Factor TheoremsThe Rational Zeros TheoremFinding the Real Zeros of a Polynomial FunctionSolving Polynomial EquationsThe Intermediate Value Theorem
2Review Topics Polynomial Long Division (pp 44-47) MAT SP 2008Review TopicsFor the next two sections, you will need to know the following:Polynomial Long Division (pp 44-47)Synthetic Division (pp 57-60)Quadratic Formula (pp )
3Review: Long Division of Polynomials MAT SP 2008Review: Long Division of PolynomialsExample
4Long Division of Polynomials MAT SP 2008Long Division of PolynomialsRemember, for every division problem the following statement is true:Dividend = (divisor) x (quotient) + remainderFor polynomial division, we express the division algorithm as:
5Long Division of Polynomials MAT SP 2008Long Division of PolynomialsUsing the previous example, we have
6Review: Synthetic Division MAT SP 2008Review: Synthetic DivisionSynthetic division is a really cool shortcut for dividing polynomials by divisors in the form x – c.As with long division, before you start synthetic division, first do the following:Arrange the terms in descending order.Use zero placeholders, where necessary.NOTE: If you are dividing by the binomial x - c, then you would use c as the divisor.Likewise, if you are dividing by x + c, then you would use (- c) as the divisor, since x + c = x - (- c).
7Synthetic Division Example #1 Divide The divisor is x – 3, so c = MAT SP 2008Synthetic DivisionExample #1DivideThe divisor is x – 3, so c =Arranging the terms of the dividend in descending order we have:So, you would set this synthetic division problem up as:cCoefficients of dividend
8Synthetic Division Example #1 (cont) Now we are ready to divide. MAT SP 2008Synthetic DivisionExample #1 (cont)Now we are ready to divide.
9Synthetic Division (continued) MAT SP 2008Synthetic Division (continued)Example #2Use synthetic division to divide:
10Quadratic FormulaTo solve equations in the form:
11The Remainder Theorem The Remainder Theorem MAT SP 2008The Remainder TheoremThe Remainder TheoremIf a polynomial f(x) is divided by x – c, the remainder r is the same as the value of f(c).This means that you can evaluate a polynomial function at a given value by substituting the value into the function,OR-using synthetic division to find the remainder.
12Exciting new synthetic method: MAT SP 2008The Remainder TheoremExampleFind f(5) given f(x) = 4x2 –10x – 21Old, boring method:Exciting new synthetic method:f(5) = 4(5)2 –10(5) – 21f(5) =
13The Factor Theorem The Factor Theorem MAT SP 2008The Factor TheoremThe Factor TheoremIf a polynomial f(c) has a factor (x – c) iff f(c) = 0.This theorem tells us that, in order for x – c to be a factor of the polynomial, the remainder when the polynomial is divided by x – c must be zero.That makes sense...For example, 8 is a factor of 32 since 8 divides into 32 evenly (the remainder is zero.)
14The Importance of the Remainder Theorem MAT SP 2008The Importance of the Remainder TheoremIf f(x) is divided by x – c and the remainder is equal to 0, then(x – c) is a factor of f(x)If c is a real number, (c, 0) is an x-intercept of the graph of f.So, from this graph of f(x), we can determine:
16The Rational Zero Theorem MAT SP 2008The Rational Zero TheoremThe Rational Zero TheoremIf the polynomial f(x)=anxn + an-1xn-1 + … + a2x2 + a1x + a0 has integer coefficients, then every rational zero must have the form , where p IS A FACTOR OF THE CONSTANT TERM a0 , and q IS A FACTOR OF THE LEADING COEFFICIENT an.This theorem will enable us to list all of the potential rational zeros of a polynomial, using the form
17Factors of the constant MAT SP 2008Factors of the constantFactors of the leading coefficient
18The Rational Zero Test Example Find all potential rational zeros of MAT SP 2008The Rational Zero TestExampleFind all potential rational zeros ofSolution
19The Rational Zero Test (continued) MAT SP 2008The Rational Zero Test (continued)Once you have the list of all potential rational zeros, you need to use trial and error to test them using synthetic division (or by substituting them into the function) to determine which ones are actual zeros. Remember, the remainder (or the functional value) must be equal to zero.A Sneaky Technology Shortcut:You can use the graph or the table of values to find one or more of the rational zeros, if there are any. Use the fact that a real zero is an x-intercept of the graph. Use this zero to perform synthetic division. A zero remainder will confirm that it is an actual zero.
20The Rational Zero Test (continued) MAT SP 2008The Rational Zero Test (continued)ExampleUse the Rational Zero Test to find ALL rational zeros of
21Using the Quotient to find the remaining zeros MAT SP 2008Using the Quotient to find the remaining zerosEach time you divide a polynomial using synthetic division, the QUOTIENT HAS A DEGREE THAT IS ONE LESS THAN THE ORIGINAL POLYNOMIAL. We will use the resulting lower-degree (“depressed”) polynomial find the remaining zeros.Your goal is obtain a lower-degree polynomial that is quadratic. Then you can find the remaining zeros by factoring or using the quadratic formula.
22Scenario 1: The original polynomial is CUBIC MAT SP 2008Scenario 1: The original polynomial is CUBICFIND ONE RATIONAL ZERO, using the Rational Zero Theorem to find potential rational zeros and then using the graph to help you locate one rational zero.DIVIDE THE CUBIC POLYNOMIAL BY THE RATIONAL ZERO using synthetic division.SOLVE THE DEPRESSED QUADRATIC EQUATION to find the remaining zeros (by factoring, completing the square, or quadratic formula).
23MAT SP 2008Example #1Find all real zeros of algebraically.
24MAT SP 2008Example #2Find all real zeros of algebraically.
25Scenario 2: The original polynomial is QUARTIC MAT SP 2008Scenario 2: The original polynomial is QUARTICFIND TWO RATIONAL ZEROS, using the rational zero test to find potential rational zeros and then using the graph to help you locate TWO rational zeros. (Be on the alert for double zeros!)DIVIDE THE QUARTIC POLYNOMIAL BY ONE OF THE RATIONAL ZEROS using synthetic division.DIVIDE THE DEPRESSED CUBIC POLYNOMIAL BY THE OTHER RATIONAL ZERO using synthetic division.SOLVE THE DEPRESSED QUADRATIC EQUATION to find the remaining zeros (by factoring, completing the square, or quadratic formula).
26MAT SP 2008Example #1Find all real zeros of algebraically.
28MAT SP 2008Example #2Show that -5 and 3 are zeros of f(x) and use this information to write the complete factorization of f.
29Approximating Zeros of Polynomial Functions MAT SP 2008Approximating Zeros of Polynomial FunctionsOften we are not able to find the zeros of a polynomial function algebraically, but we can still approximate the value of the zero.We know that the graphs of polynomials are continuous; therefore, if the sign of the function values (y-values) changes from negative to positive or vice-versa, we know that the graph must have passed through the x-axis and hence, has a zero in that interval.This result stems from the INTERMEDIATE VALUE THEOREM.
31Approximating Zeros of Polynomial Functions MAT SP 2008Approximating Zeros of Polynomial FunctionsWe can use the table of values to find the intervals (of length 1) in which a polynomial function is guaranteed to have a zero.Example:The following is a table of polynomial function values.The functional (y) values change:from + to – in the interval -4 < x < -3,from – to + in the interval –1 < x < 0,from + to – in the interval 0 < x < 1.So this polynomial function is guaranteed to have a zero in each of the following intervals: (-4, -3), (-1, 0), and (0, 1).
32Approximating Zeros of Polynomial Functions MAT SP 2008Approximating Zeros of Polynomial FunctionsExample:Use the table of values on your calculator to find the intervals (of length 1) in which the function is guaranteed to have a zero.