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Section 2-3

Real Zeros of Polynomial Functions. Section 2-3. Real Zeros of Polynomial Functions. What you should know:. 1. How to divide polynomials by other polynomials using long division. How to divide polynomial by binominals in the form (x – k) using

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Section 2-3

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  1. Real Zeros of Polynomial Functions Section 2-3

  2. Real Zerosof Polynomial Functions What you should know: 1. How to divide polynomials by other polynomials using long division. • How to divide polynomial by binominals in the form (x – k) using • synthetic division and how to recognize upper and lower bounds. 3. How to apply the Remainder and Factor Theorems. • How to determine possible rational zeros of polynomial functions using • the Rational Zero Test. • How to determine the number of positive and negative zeros a given • function has by applying Descartes Rule of Signs.

  3. Real Zeros of Polynomial Functions Long Division of Polynomials To perform long division of polynomial functions, follow the same steps as one would as if dividing a two-digit number into a three-digit number. Remember the parts to a division problem. Example: Divide 6x2 – 7x + 2

  4. Real Zeros of Polynomial Functions Try this: Label the parts. Quotient Dividend Divisor Remainder f(x) = (divisor)(quotient) + remainder Write the result as f(x) = (x + 1)(x + 2) + 3

  5. Real Zerosof Polynomial Functions The Division Algorithm If f(x) and d(x) are polynomials such that d(x)  0, and the degree of d(x) Is less than or equal to the degree of f(x), there exist unique polynomials q(x) and r(x) such that f(x) = d(x)q(x) + r(x) dividend quotient divisor remainder Where r(x) = 0 or the degree of r(x) is less than the degree of d(x). If the Remainder r(x) is zero, d(x) divides evenly into f(x) Before applying the Division Algorithm, 1. Write the dividend and divisor in descending powers of the variable. 2. Insert placeholders with zero coefficients for missing powers of the variable.

  6. Real Zerosof Polynomial Functions Example: Divide x3 +1 by x + 1. Therefore, x3 + 1 = (x + 1)(x2 - x + 1)

  7. Real Zerosof Polynomial Functions Synthetic Division (use only when dividing by divisors of the form x – k) This process requires only two mathematical operations: Addition Multiplication To divide ax3 + b x2 + c x + d by x – k, use the following process Step 1: Set x – k equal to zero and solve for k. Place this number in the window. Step 2: Beside the “window” write the coefficients of the variable in descending order along with the constant. Place a zero to hold the place of any missing power. window “the line”

  8. Real Zerosof Polynomial Functions Synthetic Division (Cont.) Step 3: Always bring down the first number in the line Step 4: Multiply the number in the window by the number “below the line” and place under the next number “above the line”. Step 5: Add ka and b and place the result below the line. Continue following Step 2until the end.

  9. Real Zerosof Polynomial Functions Example using synthetic division. Divide x4 – 10x2 – 2x + 4 by x – 3. 3 4 0 -10 -2 1 3 -15 9 -3 3 -1 -5 1 -11 To write the quotient, remember that the initial power decreases by 1. Here the quotient is x3 + 3x2 – x – 5 with a remainder of – 11. Writing it in f(x) = (divisor)(quotient) + (remainder):

  10. Real Zerosof Polynomial Functions Try these using synthetic division. Express your answer in f(x) = (divisor)(quotient) + (remainder) format. 1. 2.

  11. Real Zerosof Polynomial Functions Two Important Theorems The Remainder Theorem If a polynomial f(x) is divided by x – k, the remainder is r = f(k) This states that if f(x) is divided by x – k, the remainder will be the same as if f(x) was evaluated by f(k). Example: Remember when x4 – 10x2 – 2x + 4 was divided by x – 3 using synthetic division, the remainder was – 11. Evaluate f(x) = x4 – 10x2 – 2x + 4 by f(3). Notice that f(3) = - 11, the same as the remainder using synthetic division.

  12. Real Zerosof Polynomial Functions Try these. • Use Synthetic Division to find each of the following remainders. • Then evaluate each function using the Remainder Theorem. f(1) = 1 1. f(-2) = 4 2. f(1/2) = 4 3. f(8) = 1978 4.

  13. Real Zerosof Polynomial Functions Two Important Theorems (cont.) The Factor Theorem A polynomial f(x) has a factor of x – k if and only if f(k) =0. In the example x3 – 512 divided by x + 8, the remainder was zero. Since f( – 8) equals zero, this tells one that x + 8 is a factor and the quotient x2 – 8x + 64 is also a factor.

  14. Real Zerosof Polynomial Functions The Rational Zero Test If the polynomial f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0 Has integer coefficients, every rational zero of f has the form Rational zero = where p and q have no common factors other than 1, p is a factor of the constant term a0 and q is a factor of the leading coefficient an.

  15. Real Zerosof Polynomial Functions The Rational Zero Test (cont.) What does this mean? The Rational Zero Test gives one a list of all rational numbers which could be a zero of the given function. How does one find them? 1. List the factors of both p and q. 2. Create fractions using

  16. Real Zerosof Polynomial Functions Find all possible rational zeros of

  17. Real Zerosof Polynomial Functions Try these. Find all the possible rational zeros for the following: 1. 2.

  18. Real Zerosof Polynomial Functions Descartes Rule of Signs gives some insight on how many positive and negative zeros exist for a polynomial function. Descartes Rule of Signs Let f(x) =anxn + an-1xn-1 + … + a2x2 + a1x + a0 be a polynomial with real coefficients and a0 0. 1. The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that by an even integer. 2. The number of negative real zeros of f is either equal to the number of variations in sign of f(-x) or less than that number by an even integer.

  19. Real Zerosof Polynomial Functions Apply Descartes Rule of Signs to the following polynomial function to determine how many positive and negative real zeros may exist. Count the number of sign changes in f(x) to determine the number of possible positive real zeros. + + 1 2 3 or 1 3 Number of possible positive real zeros Less an even number

  20. Real Zerosof Polynomial Functions Applying Descartes Rule of Signs (cont.) To find the number of possible negative real zeros, f(x) must be evaluated for f(-x) and then the sign changes counted. Now count the sign changes. 0 0 0 There are no sign changes; therefore, there are no negative zeros.

  21. Real Zerosof Polynomial Functions Try these: Apply Descartes Rule of Signs to the following problems to determine how many possible positive and negative real zeros exists. 2. 1. Positive real zeros Positive real zeros: 4, 2, or 0 0 Negative real zeros: Negative real zeros: 0 3 or 1

  22. Real Zerosof Polynomial Functions Upper and Lower Bounds This is a test that can be applied while performing synthetic division to help restrict the search for zeros of a function. Let f(x) be a polynomial with real coefficients and a positive leading Coefficient. Suppose f(x) is divided by x – c, using synthetic division. • If c > 0 and each number in the last row is either positive or zero, • c is an upper bound for the real zeros of f. • If c < 0 and the numbers in the last row are alternately positive • and negative (zero entries count as positive or negative), c is a • lower bound for the real zeros of f.

  23. Real Zerosof Polynomial Functions Applying Upper and Lower Bounds The last row shows all positives. One can conclude that 1 is an upper bound and no values greater than 1 will be a zero. Example 1: The last row shows signs alternating from positive to negative. One can conclude that –1 is a lower bound and no values less than –1 will be a zero. Example 2:

  24. Real Zerosof Polynomial Functions Try these: Test to see if these are upper or lower bounds for the given function. Upper Bound Neither All positive Lower Bound Signs alternate

  25. Real Zerosof Polynomial Functions Putting it all together! Find all real zeros for Step 1: Apply Descartes Rule of Signs Positive zeros: 3 or 1 Negative zeros: 0 Step 2: Apply the Rational Zero Test Note: Since there are no negative zeros, only the positive values need to be tried in synthetic division.

  26. Real Zerosof Polynomial Functions Putting it all together! (cont.) Find all real zeros for Step 3: Apply synthetic division to find a zero. Begin with integers and look for an upper bound. All positives 2/3 is a zero because the remainder is 0. Upper Bound The resulting line is now representing a quadratic function. Set this equal to zero and solve. Try values less than 1 These are NOT real zeros.

  27. Real Zerosof Polynomial Functions Putting it all together! (finishing it up) Find all real zeros for Remember when applying the quadratic formula, 2 imaginary zeros were found. These are NOT part of the solution because only real zeros were asked for. Therefore, the only real zero for the function is

  28. Real Zerosof Polynomial Functions Wrapping it up! When finding the real zeros of a function: • Apply Descartes Rule of Signs to determine how many positive and • negative real zeros may exist. Remember to always decrease by 2 as • part of the rule. • Apply the Rational Zero Test to determine the rational possibilities for • the zeros. • Use synthetic division to find zeros. Look for upper and lower bounds • to possibly eliminate some of the rational values given by the Rational • Zero Test. • Once you have found a zero, determine if the resulting line from synthetic • division is a quadratic. If so, write the quadratic equation and solve. If not, • continue with synthetic division always using the resulting line to find another • zero. Try the exact same value in the window again to see if a double root • exists. If not, try another possibility from the Rational Zero Test.

  29. Real Zerosof Polynomial Functions Try this. Find all real zeros for Step 1: Descartes Rule of Signs Positive: 1 2 or 0 Negative: Step 2: Rational Zero Test Step 3: Find the zeros using Synthetic Division and/or solve a quadratic equation -2, -1, and 2 Real Zeros are:

  30. Real Zerosof Polynomial Functions What you should know: 1. How to divide polynomials by other polynomials using long division. • How to divide polynomial by binominals in the form (x – k) using • synthetic division and how to recognize upper and lower bounds. 3. How to apply the Remainder and Factor Theorems. • How to determine possible rational zeros of polynomial functions using • the Rational Zero Test. • How to determine the number of positive and negative zeros a given • function has by applying Descartes Rule of Signs.

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