PreparationChapter4

PreparationChapter4 - Preparation for Chapter 4 1-...

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Preparation for Chapter 4 1- Synthetic Division If a polynomial p(x) is divided by (x – c) we can apply synthetic division: Let’s divide x 3 – 4x 2 – 5 by x – 3 Step 1: Write the dividend in descending powers of x. Copy the coefficients; write a zero for any missing power. 1 -4 0 -5 Step 2: Create the following structure: 3 1 ! 4 0 ! 5 Row 1 Row 2 Row 3 First row, we place: 1 -4 0 -5 Second row, to the left of the symbol, we place 3 since the divisor is x – 3. Step 3: Bring the 1 down two rows, and enter it in row 3. 3 1 ! 4 0 ! 5 " 1 Step 4: Multiply the 1, in row 3, by 3, and place the result 3, in row 2 underneath the -4. 3 1 ! 4 0 ! 5 " 3 1 # 3 ! Step 5: Add the entries in row 1 and row 2 (-4 + 3 = -1) and enter the sum in row 3. 3 1 ! 4 0 ! 5 " 3 1 # 3 ! ! 1 Step 6: Multiply the -1, in row 3, by 3, and place the result -3, in row 2 underneath the 0. 3 1 ! 4 0 ! 5 " 3 ! 3 1 # 3 ! ! 1 # 3 ! Step 7: Add the entries in row 1 and row 2 (0 + -3 = -3) and enter the sum in row 3. 3 1 ! 4 0 ! 5 " 3 ! 3 1 # 3 ! ! 1 # 3 ! ! 3
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3 1 ! 4 0 ! 5 " 3 ! 3 ! 9 1 # 3 ! ! 1 # 3 ! ! 3 # 3 ! Step 9: add the entries in row 1 and row 2 (-5 + -9 = -14) and enter the sum in row 3. 3 1 ! 4 0 ! 5 " 3 ! 3 ! 9 1 # 3 ! ! 1 # 3 ! ! 3 # 3 ! ! 14 End of the process. The numbers 1 -1 and -3 are the coefficients of a second degree polynomial that is the quotient (x 2 – x – 3), the number -14 is the remainder, so x 3 - 4x 2 – 5 = (x - 3)(x 2 – x – 3) + (-14) Example : Use synthetic division to divide (2x 5 + 5x 4 – 2x 3 + 2x 2 – 2x + 3) by (x + 3) The divisor is x + 3 = x – (-3), so we place -3 to the left of the symbol. Remember, we add each entry in row 1 to the corresponding entry in row 2 and the sum is place in row 3. ! 3 2 5 ! 2 2 ! 2 3 Row 1 " ! 6 3 ! 3 3 ! 3 Row 2 2 # ! 3 ( ) ! ! 1 # ! 3 ( ) ! 1 # ! 3 ( ) ! ! 1 # ! 3 ( ) ! 1 # ! 3 ( ) ! 0 Row 3 The remainder is 0, the quotient is the polynomial 2x 4 – x 3 + x 2 – x + 1, so 2x 5 + 5x 4 – 2x 3 + 2x 2 – 2x + 3) = (x + 3)( 2x 4 – x 3 + x 2 – x + 1) 2- Roots or Zeros of a Polynomial Given a polynomial P(x), we say that a number c is a root or zero of the polynomial if P(c) = 0. The Factor Theorem : The number c is a root or zero of the polynomial P(x) if and only if (x – c) is a factor of P(x). (P(x) = (x – c) Q(x) where degree Q(x) equals degree P(x) minus 1) Example : Show that x – 1 is a factor of P(x) = 2x 3 – x 2 + 2x – 3. Applying the theorem, we have to show that P(1) = 0. P(1) = 2(1)
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PreparationChapter4 - Preparation for Chapter 4 1-...

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