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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:. 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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  1. Using the Fundamental Theorem of Algebra!!! 6.7 Pg.366 This ppt includes 7 slides consisting of a Review and 3 examples

  2. 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.

  3. 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

  4. 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

  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

  6. 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.

  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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