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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

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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 - PowerPoint PPT Presentation

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Page 1: Using the Fundamental Theorem of Algebra!!!

USING THE FUNDAMENTAL

THEOREM OF ALGEBRA!!!

6.7Pg.366

This ppt includes 7 slides consisting of a Review and3 examples

Page 2: Using the Fundamental Theorem of Algebra!!!

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

2 2

A polynomial to the nth degree will have n zeros.

Page 3: Using the Fundamental Theorem of Algebra!!!

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 = -16x = ±√-16 = ±4i

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

Page 4: 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

Page 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

Page 6: Using the Fundamental Theorem of Algebra!!!

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.

Page 7: Using the Fundamental Theorem of Algebra!!!

THIS ENDS CHAPTER 6.7 Assignments will be made in class and

placed on the web page under lesson plans.


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