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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

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 3 16 48
  • -3 -3 0 -48

1 0 16 0

x2 + 16 = 0

x2 = -16

x = ±√-16 = ±4i

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

slide4
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
slide5
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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