Zeros of Polynomial Functions


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1 Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction Polynomial 5x 3 + 3x 2 + (2 + 4i) + i 5x 3 + 3x 2 + 2x π Type of Coefficient complex real Objective: To find a polynomial with specified zeros, rational zeros, and other zeros. 5x 3 + 3x 2 + ½ x ⅜ 5x 3 + 3x 2 + 8x 11 rational integer Rational Zero Theorem If the polynomial f(x) = a n x n + a n1 x n a 1 x + a 0 has integer coefficients, then every rational of f(x) is of the form p q where p is a factor of the coefficient a 0 and q is a factor of the coefficient a n. Rational Root (Zero) Theorem (in other words) If q is the leading coefficient and p is the constant term of a polynomial, then the only possible rational roots are + factors of p divided by + factors of q. (p / q) 1
2 Rational Root (Zero) Theorem (in other words) 5 3 : f ( x) = 6x 4x 12x + 4 To find the POSSIBLE rational roots of f(x), we need the FACTORS of the leading coefficient (6 for this example) and the factors of the constant term (4, for this example). Possible rational roots are ± factors of p ± 1, ± 2, ± = = ± 1, 2, 4,,,,, ± factors of q ± 1, ± 2, ± 3, ± List all possible rational zeros of f(x) = x 3 + 2x 2 5x 6. Another example List all possible rational zeros of f(x) = 4x x 4 x 3. How do we know which possibilities are really zeros (solutions)? Use trial and error and division to see if one of the possible zeros is actually a zero. Remember: When dividing by x c, if the is 0 when using synthetic division, then c is a zero of the polynomial. If c is a zero, then solve the polynomial resulting from the synthetic division to find the other zeros. Find all zeros of f(x) = x 3 + 8x x 20. Finding the Rational Zeros of a Polynomial 1. List all rational zeros of the polynomial using the Rational Zero Theorem. 2. Use synthetic division on each possible rational zero and the polynomial until one gives a remainder of. This means you have found a zero, as well as a factor. 3. Write the polynomial as the of this factor and the quotient. 4. Repeat procedure on the quotient until the quotient is 5. Once the quotient is quadratic, factor or use the quadratic formula to find the remaining real and imaginary zeros. 2
3 Find all zeros of f(x) = x 3 + x 25x 2. List all possible zeros, and use synthetic division to test and find an actual zero. Then use the quotient to find the remaining zeros. f(x) = x 3 4x 2 + 8x  5 More review  List all possible zeros. Use synthetic division to test and find an actual zero. Then use the resulting quotient to find the remaining zeros. f(x) = x 3 + 4x 23x  6 How many zeros, not necessarily rational, does a polynomial with rational coefficients have? An nth degree polynomial has a total of n. Some may be rational, irrational or complex. Because all coefficients are RATIONAL, irrational roots exist in (both the irrational # and its conjugate). roots also exist in pairs (both the complex # and its conjugate). If a + bi is a root, a bi is a root If a + b is a root, a b is a root. NOTE: Sometimes it is helpful to graph the function and find the xintercepts (zeros) to narrow down all the possible zeros. Solve: x 46x x 230x + 13 = 0. Fundamental Theorem of Algebra If f(x) is a polynomial function of degree n, where n > 1, then the equation f(x) = 0 has at least one complex zero, real or imaginary. Note: This theorem just guarantees a zero exists, but does not tell us how to find it. 3
4 Linear Factorization Theorem Remember Complex zeros come in pairs as complex conjugates: a + bi, a bi Irrational zeros come in pairs. a+ c b, a c b More Practice Find a polynomial function, in factored form, of degree 5 with 1/2 as a zero with multiplicity 2, 0 as a zero of multiplicity 1, and 1 as a zero of multiplicity 2. Practice Find a polynomial function of degree 3 with 2 and i as zeros. Find a thirddegree polynomial function f(x) with real coefficients that has 3 and i as zeros and such that f(1) = 8. Solve the given polynomial equation. Use the Rational Zero Theorem, or graph as an aid to obtaining the first zero. x 4 x 3 + 2x 2 4x 8 = 0. 4
5 Extra Suppose that a polynomial function of degree 4 with rational coefficients has i and (3 + 3) as zeros. Find the other zero(s). Extra Find a polynomial of degree 3 where 4 and 2i are zeros, and f(1) = 50. Extra Use the Rational Zero Theorem to list all the possible zeros for f(x) = 4x 5 8x 4 x
Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
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