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Learning Objectives

Aim 1: How do we divide a polynomial by a polynomial using long division? synthetic division? Aim 2: How do the remainder and factor theorems help us find roots/zeros of polynomials?. Learning Objectives. SWBAT

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Learning Objectives

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  1. Aim 1: How do we divide a polynomial by a polynomial using long division? synthetic division?Aim 2: How do the remainder and factor theorems help us find roots/zeros of polynomials?

  2. Learning Objectives • SWBAT • Divide a polynomial by another polynomial and then rewrite using the algorithm p(x) = d(x)q(x)+r(x) • Use synthetic division to simplify division of polynomials. • Rewrite quotients obtained in synthetic division in polynomial/remainder form • Use the factor theorem and remainder theorem to find roots and evaluate polynomials.

  3. Long Division - divide a polynomial by a polynomial • Think back to long division from 3rd grade. • How many times does the divisor go into the dividend? Put that number on top. • Multiply that number by the divisor and put the result under the dividend. • Subtract and bring down the next number in the dividend. Repeat until you have used all the numbers in the dividend.

  4. x2/x = x -8x/x = -8 x - 8 + 3x -( ) x2 - 8x - 24 -( ) - 8x - 24 0

  5. h3/h = h2 4h2/h = 4h 5h/h = 5 h2 + 4h + 5 -( ) - 4h2 h3 - 11h 4h2 -( ) 4h2 - 16h 5h + 28 -( ) 5h - 20 48

  6. Synthetic Division - divide a polynomial by a polynomial • To use synthetic division: • There must be a coefficient for every possible power of the variable. • The divisor must have a leading coefficient of 1.

  7. Since the numerator does not contain all the powers of x, you must include a 0 for the Step #1: Write the terms of the polynomial so the degrees are in descending order.

  8. 5 0 -4 1 6 Step #2: Write the constant r of the divisor x-r to the left and write down the coefficients. Since the divisor is x-3, r=3

  9. 5 Step #3: Bring down the first coefficient, 5.

  10. Step #4: Multiply the first coefficient by r, so and place under the second coefficient then add. 15 5 15

  11. 15 45 15 5 Step #5: Repeat process multiplying the sum, 15, by r; and place this number under the next coefficient, then add. 41

  12. 15 45 123 372 15 41 5 Step #5 cont.: Repeat the same procedure. Where did 123 and 372 come from? 124 378

  13. 15 45 123 372 15 41 124 378 5 Step #6: Write the quotient. The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.

  14. The quotient is: Remember to place the remainder over the divisor.

  15. Ex 7: Step#1: Powers are all accounted for and in descending order. Step#2: Identify r in the divisor. Since the divisor is x+4, r=-4 .

  16. Step#3: Bring down the 1st coefficient. Step#4: Multiply and add. Step#5: Repeat. 4 -4 20 0 8 -1 1 0 -2 10 -5

  17. Ex 8: Notice the leading coefficient of the divisor is 2 not 1. We must divide everything by 2 to change the coefficient to a 1.

  18. 3

  19. *Remember we cannot have complex fractions - we must simplify.

  20. Ex 9: Coefficients 1

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