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1 5.5 Real Zeros of Polynomial Functions In this section, we will study the following topics: The Remainder and Factor Theorems The Rational Zeros Theorem Finding the Real Zeros of a Polynomial Function Solving Polynomial Equations The Intermediate Value Theorem

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2 Review Topics For the next two sections, you will need to know the following: 1. Polynomial Long Division (pp 44-47) 2. Synthetic Division (pp 57-60) 3. Quadratic Formula (pp 102-104)

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3 Example Review: Long Division of Polynomials

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4 Remember, for every division problem the following statement is true: Dividend = (divisor) x (quotient) + remainder Long Division of Polynomials For polynomial division, we express the division algorithm as:

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5 Long Division of Polynomials Using the previous example, we have

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6 Synthetic 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: 1. Arrange the terms in descending order. 2. 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). Review: Synthetic Division

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7 Synthetic Division Example #1 Divide The 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: c Coefficients of dividend

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8 Synthetic Division Example #1 (cont) Now we are ready to divide.

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9 Synthetic Division (continued) Example #2 Use synthetic division to divide:

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10 Quadratic Formula To solve equations in the form:

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11 The Remainder Theorem 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. The Remainder Theorem If a polynomial f(x) is divided by x – c, the remainder r is the same as the value of f(c).

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12 The Remainder Theorem Example Find f(5) given f(x) = 4x 2 –10x – 21 Old, boring method: synthetic Exciting new synthetic method: f(5) = 4(5) 2 –10(5) – 21 f(5) =

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13 The Factor Theorem If 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.)

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14 The Importance of the Remainder Theorem If 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:

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16 The Rational Zero Theorem If the polynomial f(x)=a n x n + a n-1 x n-1 + … + a 2 x 2 + a 1 x + a 0 has integer coefficients, then every rational zero must have the form, where p IS A FACTOR OF THE CONSTANT TERM a 0, and q IS A FACTOR OF THE LEADING COEFFICIENT a n. This theorem will enable us to list all of the potential rational zeros of a polynomial, using the form

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Factors of the constant Factors of the leading coefficient

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18 The Rational Zero Test Example Find all potential rational zeros of Solution

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19 The 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.

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20 The Rational Zero Test (continued) Example Use the Rational Zero Test to find ALL rational zeros of

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21 Using the Quotient to find the remaining zeros Each 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.

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22 FIND 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). Scenario 1: The original polynomial is CUBIC

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23 Find all real zeros of algebraically. Example #1

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24 Find all real zeros of algebraically. Example #2

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25 FIND 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). Scenario 2: The original polynomial is QUARTIC

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26 Find all real zeros of algebraically. Example #1

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28 Show that -5 and 3 are zeros of f(x) and use this information to write the complete factorization of f. Example #2

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29 Approximating Zeros of Polynomial Functions Often 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.

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31 Approximating Zeros of Polynomial Functions We can use the table of values to find the intervals (of length 1) in which a polynomial function is guaranteed to have a zero. 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). Example: The following is a table of polynomial function values.

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32 Approximating Zeros of Polynomial Functions Example: 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.

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33 End of Sect. 5.5

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