academy algebra ii/trig 5.5: the real zeros of a polynomial functions hw: p.387 (14, 27, 30, 31, 37,...

Academy Algebra II/TrigAcademy Algebra II/Trig

5.5: The Real Zeros of a 5.5: The Real Zeros of a Polynomial FunctionsPolynomial Functions

HW: p.387 (14, 27, 30, 31, HW: p.387 (14, 27, 30, 31, 37, 38, 46, 51)37, 38, 46, 51)

Factor Theorem: Factor Theorem: If f(c) = 0, then x-c is a factor of f(x).If f(c) = 0, then x-c is a factor of f(x).

Using the factor theorem, determine Using the factor theorem, determine whether the function has the factor given.whether the function has the factor given.

1;322)( 23 xxxxxf

Note: You could also divide to check.

List all possible rational zeros using the List all possible rational zeros using the rational zero theorem.rational zero theorem.

• Let f be a polynomial of degree 1 or higher of the form:– The possible rational zeros =

na

a

q

p

t coefficien leading offactor

ermconstant t offactor 0

011

1 ...)( axaxaxaxf nn

nn


• Example: List the possible rational zeros for the function: 12112)( 23 xxxxf

1

12,

1

6,

1

4,

1

3,

1

2,

1

1Possible rational zeros:

Possible rational zeros: 12,6,4,3,2,1

Factors of the constant:

Factors of the leading coefficient:

12,6,4,3,2,1

1


10934)( .)1 234 xxxxxf

61132)( .)2 23 xxxxf

Descartes’ Rule of SignsDescartes’ Rule of Signs

• # of positive real zeros: # of sign changes for f(x) or an even integer less than the sign change.

• # of negative real zeros: # of sign changes for f(-x) or an even integer less than the sign change.

Use Descartes’ Rule of Signs to discuss Use Descartes’ Rule of Signs to discuss the real zeros of the polynomial.the real zeros of the polynomial.

10934)( .)1 234 xxxxxf

61132)( .)2 23 xxxxf

Find the zeros of a polynomial function.Find the zeros of a polynomial function.

• List the possible rational zeros of the function.• Use Descartes’ Rule of Signs to determine the

possible number of positive and negative zeros.

• Test the zeros using division. (Since the zeros are x-intercepts, when you divide you should end up with a remainder of zero.)– Factor if you get a factorable polynomial.

Find all real zeros of the function.Find all real zeros of the function.

18154)( 23 xxxxf


163224125)( 2345 xxxxxxf


127421110)( 234 xxxxxf


5.5: The Real Zeros of a Polynomial 5.5: The Real Zeros of a Polynomial FunctionsFunctions

HW: p.387-388 (55-58, 67, 70)HW: p.387-388 (55-58, 67, 70)

Test: 4.3-4.5, 5.1, 5.5, 5.6Test: 4.3-4.5, 5.1, 5.5, 5.6

Find the real solutions of the equation.Find the real solutions of the equation.

02743 23 xxx

Find the real solutions of the equation.Find the real solutions of the equation.

0624 234 xxxx


5.6: Complex Zeros, Fundamental 5.6: Complex Zeros, Fundamental Theorem of AlgebraTheorem of Algebra

Find the zeros of the Find the zeros of the polynomial function.polynomial function.

f(x) = 3x4 + 5x3 + 25x2 + 45x – 18

Do Now: Solve the equation.Do Now: Solve the equation.x3 + 5x2 + 4x + 20 = 0


5.6: Complex Zeros, Fundamental 5.6: Complex Zeros, Fundamental Theorem of AlgebraTheorem of Algebra

HW: p.394 (16, 20, 21, 31-33, 40)HW: p.394 (16, 20, 21, 31-33, 40)

Test: 4.3-4.5, 5.1, 5.5, 5.6Test: 4.3-4.5, 5.1, 5.5, 5.6

Fundamental Theorem of AlgebraFundamental Theorem of Algebra

• If f(x) is a polynomial with degree of n (where n>0), then the equation f(x) = 0 has at least one solution.

• Corollary: The equation f(x) = 0 has exactly n solutions provided you count multiple solutions.

Conjugates TheoremConjugates Theorem• If is an imaginary zero of

function f, then is also a zero.• If is a zero of function f where

is irrational, then is also a zero. ba

ba b

bia bia

Write a polynomial function f of least Write a polynomial function f of least degree that has rational coefficients, a degree that has rational coefficients, a

leading coefficient of 1, & the given zeros.leading coefficient of 1, & the given zeros.1.) -1, 2, 4

2.) 4, 2i, 51

Write a polynomial function f with the Write a polynomial function f with the given degree, rational coefficients, a given degree, rational coefficients, a

leading coefficient of 1, & the given zeros.leading coefficient of 1, & the given zeros.

3.) degree 4; zeros: 3, multiplicity 2; i

Find all zeros of the Find all zeros of the polynomial function.polynomial function.

f(x) = x4 - 13x2 + 36

academy algebra ii/trig 5.5: the real zeros of a polynomial functions hw: p.387 (14, 27, 30, 31, 37,...

Documents