Finding Zeros of Polynomials Last updated: 12-4-07

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Divide:

Use Synthetic Division to Divide x + 2 = 0 1 4 -5 3 -2 -4 18 2 -9 21 x = -2 remainder

Use Synthetic Division to Divide x - 3 = 0 2 -11 3 36 6 -15 -36 -5 -12 x = 3 Factored Form:

Factor x + 1 = 0 1 4 -15 -18 -1 -3 18 3 x = -1

Find the zeros of x = 3 x – 3 = 0 6 -7 -43 30 3 18 33 -30 11 -10

A polynomial f(x) has a factor x – k if and only if f(k) = 0. Factor Theorem A polynomial f(x) has a factor x – k if and only if f(k) = 0.

Rational Zero Theorem If f(x) = anxn + . . . + a1x + a0 has integer coefficients, then every rational zero of f(x) has the following form: p = factor of constant term a0 q factor of leading coefficient an

List the possible rational zeros 15 p = factor of constant term a0 q factor of leading coefficient an 1, 3, 5, 15 6 1, 2, 3, 6

List the possible rational zeros 24 p = factor of constant term a0 q factor of leading coefficient an 1, 2, 3, 4, 6, 8, 12, 24 9 1, 3, 9

Find all real zeros 3 p = factor of constant term a0 q factor of leading coefficient an 1, 3 8 1, 2, 4, 8

Find all real zeros 8 2 -21 -7 3 1 10 -11 -18 -15 x = 1 Remainder ≠ 0 Therefore, not a factor.

Find all real zeros 8 2 -21 -7 3 24 78 171 492 26 57 164 495 x = 3 Remainder ≠ 0 Therefore, not a factor.

Find all real zeros 8 2 -21 -7 3 -12 15 9 -3 -10 -6

Find all real zeros 1 3 = 1, 3 8 1, 2, 4, 8 4

Find all real zeros 4 -5 -3 1 -1 -4

Find all real zeros Rational Rational Irrational but Real

Look at the graph ≈ 1.618 ≈ -0.618

Look at the graph

Find all real zeros 25 p = factor of constant term a0 q factor of leading coefficient an 1, 5, 25 1 1

Find all real zeros 1 -16 -40 -25 -15 -55 -80 5 25 45 9 x = 1 x = 5 -16 -40 -25 -15 -55 -80 5 25 45 9 x = 5 x – 5 = 0

Find all real zeros x = -1 1 5 9 -1 -4 -5 4 x + 1 = 0

Find all real zeros Imaginary -- not Real

Look at the graph Note: x-min: -10 x-max: 10 x-scale: 1 y-min: -250 y-max: 100 y-scale: 50

Find all real zeros 12 p = factor of constant term a0 q factor of leading coefficient an 1, 2, 3, 4, 6, 12 1 1

Find all real zeros x = 1 1 -3 -5 15 4 -12 -2 -7 8 12 x - 1 = 0

Find all real zeros x = 2 1 -2 -7 8 12 2 -14 -12 -6 x - 2 = 0

Find all real zeros x = 3 1 -7 -6 3 9 6 2 x - 3 = 0

Find all real zeros

Look at the graph End Behavior?

Look at the graph Note: x-min: -5 x-max: 5 x-scale: 1 y-min: -20 y-max: 20 y-scale: 5

Fundamental Theorem of Algebra If f(x) is a polynomial function of degree n where n>0, then the equation f(x) = 0 has at least one solution in the set of complex numbers.

Corollary to the Fundamental Theorem of Algebra If f(x) is a polynomial function of degree n where n>0, then the equation f(x) = 0 has exactly n solutions provided each solution repeated twice is counted as 2 solutions, each solution repeated three times is counted as 3 solutions, and so on.

Find all real zeros 9 p = factor of constant term a0 q factor of leading coefficient an 1, 3, 9 1 1

Find all real zeros 1 is a multiple root with multiplicity 3

Find all real zeros 20 p = factor of constant term a0 q factor of leading coefficient an 1, 2, 4, 5, 10, 20 1 1

Find all real zeros 1 -6 7 16 -18 -20 -5 2 18 -8 -2 28 20 -4 -1 14 10 -8 -2 28 20 -4 -1 14 10 x = 1 x = 2 x - 2 = 0

Find all real zeros 1 -4 -1 14 10 2 -10 8 -2 -5 4 18 5 x = 2 x = -1 x = 2 x = -1 x + 1 = 0

Find all real zeros 1 -5 4 10 -1 6 -10 -6 x = -1 x + 1 = 0

Find all real zeros

End Behavior?

Imaginary solutions appear in conjugate pairs. Key Concepts If f(x) is a polynomial function with real coefficients, and a + bi is an imaginary zero of f(x), then a - bi is also a zero of f(x). Imaginary solutions appear in conjugate pairs.

Key Concepts If f(x) is a polynomial function with rational coefficients, and a and b are rational numbers such that ---- is irrational. If --------- is a zero of f(x), then --------- is also a zero of f(x). Irrational solutions containing a square root appear in conjugate pairs.

f(x) = (x – 1) (x + 2) (x – 4) f(x) = (x – 1) (x2 – 4x + 2x – 8) Write a polynomial function f(x) of least degree that has rational coefficients, leading coefficient of 1, and the following zeros  1, -2, 4. x = 1 x = -2 x = 4 x – 1 = 0 x + 2 = 0 x – 4 = 0 f(x) = (x – 1) (x + 2) (x – 4) f(x) = (x – 1) (x2 – 4x + 2x – 8) f(x) = (x – 1) (x2 – 2x – 8) f(x) = x3 – 2x2 – 8x – x2 + 2x + 8 f(x) = x3 – 3x2 – 6x + 8

Write a polynomial function f(x) of least degree that has rational coefficients, leading coefficient of 1, and the following zeros  WAIT !

Write a polynomial function f(x) of least degree that has rational coefficients, leading coefficient of 1, and the following zeros 

Write a polynomial function f(x) of least degree that has rational coefficients, leading coefficient of 1, and the following zeros  WAIT !

Write a polynomial function f(x) of least degree that has rational coefficients, leading coefficient of 1, and the following zeros 

Descartes’ Rule of Signs Let f(x) = anxn + an-1xn-1 +. . . + a1x + a0 be a polynomial function with real coefficients.  The number of positive real zeros of f(x) is equal to the number of changes in sign of the coefficients of f(x) or is less than this by an even number.  The number of negative real zeros of f(x) is equal to the number of changes in sign of the coefficients of f(-x) or is less than this by an even number.

Determine the number of positive real zeros. 1 2 3 3 sign changes  f(x) has 3 or 1 positive real zero(s).

Determine the number of negative real zeros. 1 2 2 sign changes  f(x) has 2 or 0 negative real zero(s).

Putting it together !  f(x) has 3 or 1 positive real zero(s)  f(x) has 2 or 0 negative real zero(s) Positive real zeros Negative real zeros Imaginary zeros Total zeros 3 2 5 1 4

Look at the graph Note: x-min: -5 x-max: 5 x-scale: 1 y-min: -30 y-max: 20 y-scale: 5

Determine the number of positive real zeros. 0 sign changes  f(x) has 0 positive real zero(s).

Determine the number of negative real zeros. 1 2 3 3 sign changes  f(x) has 3 or 1 negative real zero(s).

Putting it together !  f(x) has 0 positive real zero(s)  f(x) has 3 or 1 negative real zero(s) Positive real zeros Negative real zeros Imaginary zeros Total zeros 3 4 7 1 6

Look at the graph Note: x-min: -5 x-max: 5 x-scale: 1 y-min: -50 y-max: 50 y-scale: 10

Find all real zeros 21 p = factor of constant term a0 q factor of leading coefficient an 1, 3, 7, 21 18 1, 2, 3, 6, 9, 18  f(x) has 2 or 0 positive real zero(s)  f(x) has 1 negative real zero(s)

Find all real zeros

Find all real zeros upper bound 18 -63 40 21 1 -45 -5 16 2 36 -54 -28 -27 -14 -7 3 54 39 -9 13 60 7 126 441 3367 63 481 3388 upper bound

Find all real zeros X X X X X

Find all real zeros upper bound 18 -63 40 21 1 -45 -5 16 2 36 -54 -28 Location Principle 18 -63 40 21 1 -45 -5 16 2 36 -54 -28 -27 -14 -7 3 54 39 -9 13 60 7 126 441 3367 63 481 3388 upper bound

Find all real zeros X X X X X X X X X X X X

Find all real zeros 18 -63 40 21 27 -54 -21 -36 -14 42 14 6