do now: find all real zeros of the function.. academy algebra ii 5.7: apply the fundamental theorem...

Do Now: Find all real zeros of the Do Now: Find all real zeros of the function. function. 42 19 18 5 2 ) ( 2 3 4 x x x x x f

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Page 1: Do Now: Find all real zeros of the function.. Academy Algebra II 5.7: Apply the Fundamental Theorem of Algebra HW Friday: p.384 (12-18 even) Test 5.5-5.9:

Do Now: Find all real zeros of the function.Do Now: Find all real zeros of the function.

42191852)( 234 xxxxxf

Page 2: Do Now: Find all real zeros of the function.. Academy Algebra II 5.7: Apply the Fundamental Theorem of Algebra HW Friday: p.384 (12-18 even) Test 5.5-5.9:

Academy Algebra IIAcademy Algebra II

5.7: Apply the Fundamental 5.7: Apply the Fundamental

Theorem of AlgebraTheorem of Algebra

HW Friday: p.384 (12-18 even)HW Friday: p.384 (12-18 even)

Test 5.5-5.9: Tuesday, 12/2Test 5.5-5.9: Tuesday, 12/2

Page 3: Do Now: Find all real zeros of the function.. Academy Algebra II 5.7: Apply the Fundamental Theorem of Algebra HW Friday: p.384 (12-18 even) Test 5.5-5.9:

Descartes’ Rule of SignsDescartes’ Rule of Signs• The number of positive real zeros of f is

equal to the number of sign changes in the sign of the coefficients of f(x) or is less than this by an even number.

• The number of negative real zeros of f is equal to the number of changes in sign of the coefficients of f(-x) or is less than this by an even number.

Page 4: Do Now: Find all real zeros of the function.. Academy Algebra II 5.7: Apply the Fundamental Theorem of Algebra HW Friday: p.384 (12-18 even) Test 5.5-5.9:

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 each solution repeated twice is counted as 2 solutions, each solution repeated three times is counted as 3 solutions, and so on.

Page 5: Do Now: Find all real zeros of the function.. Academy Algebra II 5.7: Apply the Fundamental Theorem of Algebra HW Friday: p.384 (12-18 even) Test 5.5-5.9:

Fundamental Theorem of AlgebraFundamental Theorem of Algebra• Example: x3 – 5x2 – 8x + 48 = 0

(x+3)(x – 4)2= 0

x = -3 and x = 4

This equation only has two distinct solutions: -3 and 4. Because the factor (x – 4) appears twice, you can count the solution 4 twice. With the repeated solution the third-degree equation has three solutions.

Page 6: Do Now: Find all real zeros of the function.. Academy Algebra II 5.7: Apply the Fundamental Theorem of Algebra HW Friday: p.384 (12-18 even) Test 5.5-5.9:

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

Page 7: Do Now: Find all real zeros of the function.. Academy Algebra II 5.7: Apply the Fundamental Theorem of Algebra HW Friday: p.384 (12-18 even) Test 5.5-5.9:

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

f(x) = x4 – 8x3 + 18x2 – 27

Page 8: Do Now: Find all real zeros of the function.. Academy Algebra II 5.7: Apply the Fundamental Theorem of Algebra HW Friday: p.384 (12-18 even) Test 5.5-5.9:

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

f(x) = x5 – 2x4 + 8x2 – 13x + 6

Page 9: Do Now: Find all real zeros of the function.. Academy Algebra II 5.7: Apply the Fundamental Theorem of Algebra HW Friday: p.384 (12-18 even) Test 5.5-5.9:

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

Page 10: Do Now: Find all real zeros of the function.. Academy Algebra II 5.7: Apply the Fundamental Theorem of Algebra HW Friday: p.384 (12-18 even) Test 5.5-5.9:

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, 51

Page 11: Do Now: Find all real zeros of the function.. Academy Algebra II 5.7: Apply the Fundamental Theorem of Algebra HW Friday: p.384 (12-18 even) Test 5.5-5.9:

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.3.) 2, 2i, 64

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