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Using the Fundamental Theorem of Algebra!!!PowerPoint Presentation

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)=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 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

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

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. real coefficients, a leading coeff. of 1 and 4, 4, 2+i as 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 real coefficients, a leading coeff. of 1 and 4, 4, 2+i as zeros.

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

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