Using the fundamental theorem of algebra
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Using the Fundamental Theorem of Algebra!!!. 6.7 Pg.366 This ppt includes 7 slides consisting of a Review and 3 examples. Review:. Find all the zeros: f (x)=x 3 +x 2 -2x-2 Answer: - , ,-1 F(x)= x 3 – 6x 2 – 15 x + 100 = (x + 4)(x – 5)(x – 5) the zeros are: -4, 5, 5

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Using the Fundamental Theorem of Algebra!!!

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Using the fundamental theorem of algebra

Using the Fundamental Theorem of Algebra!!!

6.7

Pg.366

This ppt includes 7 slides consisting of a Review and

3 examples


Review

Review:

  • Find all the zeros:

  • f(x)=x3+x2-2x-2

  • Answer: - , ,-1

  • F(x)= x3 – 6x2 – 15 x + 100 = (x + 4)(x – 5)(x – 5)

  • the zeros are: -4, 5, 5

  • 5 is a repeated solution

A polynomial to the nth degree will have n zeros.


Example find all zeros of x 3 3x 2 16x 48 0

Example: find all zeros of x3 + 3x2 +16x +48= 0

  • (should be 3 total! degree 3)

  • CT = ±1 ±2 ±3 ±4 ±6 ±8 ±12 ±16 ±24 ±48

  • LC±1

  • Graph the equation and you’ll see only 1 real zero:

  • Look in the table and you will find -3 is the only zero in the table, SO use synthetic division with -3

  • 1 31648

  • -3 -30 -48

    1 0 160

    x2 + 16 = 0

    x2 = -16

    x = ±√-16 = ±4i

    The three zeros are -3, 4i, -4i


Using the fundamental theorem of algebra

Now write a polynomial function of least degree that has real coefficients, a leading coefficint of 1 and 1, -2+i, -2-i as zeros.

  • F(x)= (x-1)(x-(-2+i))(x-(-2-i))

  • F(x)= (x-1)(x+2-i)(x+2+i)

  • f(x)= (x-1){(x+2)-i} {(x+2)+i}

  • F(x)= (x-1){(x+2)2-i2} Foil

  • F(x)=(x-1)(x2 + 4x + 4 –(-1))Take care of i2

  • F(x)= (x-1)(x2 + 4x + 4 + 1)

  • F(x)= (x-1)(x2 + 4x + 5)Multiply

  • F(x)= x3 + 4x2 + 5x – x2 – 4x – 5

  • f(x)= x3 + 3x2 + x - 5


Using the fundamental theorem of algebra

Now write a polynomial function of least degree that has real coefficients, a leading coeff. of 1 and 4, 4, 2+i as zeros.

  • Note: 2+i means 2-i is also a zero

  • F(x)= (x-4)(x-4)(x-(2+i))(x-(2-i))

  • F(x)= (x-4)(x-4)(x-2-i)(x-2+i)

  • F(x)= (x2 – 8x +16)((x-2)-i)((x-2)+i)

  • F(x)= (x2 – 8x +16)((x-2)2-i2)

  • F(x)= (x2 – 8x +16)(x2 – 4x + 4 –(-1))

  • F(x)= (x2 – 8x +16)(x2 - 4x + 5)

  • F(x)= x4–4x3+5x2–8x3+32x2-40x+16x2-64x+80

  • F(x)= x4-12x3+53x2-104x+80


Using a graphing calculator to find the real zeros

Using a graphing calculator to find the real zeros.

  • Under y= type in the equation.

  • Go to second; calc; 2:zero

  • Left bound: you need to place the cursor to the left of the intersection and press enter.

  • Right bound: you need to place the cursor to the right of the intersection and press enter; and enter again.

  • At the bottom of the window “zero” will appear x = # This is your real zero.


This ends chapter 6 7

This ends Chapter 6.7

  • Assignments will be made in class and placed on the web page under lesson plans.


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