# Using the Fundamental Theorem of Algebra!!! - PowerPoint PPT Presentation

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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!!!

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

• 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 31648

• -3 -30 -48

1 0 160

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.

• 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

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