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Lesson 2.3 Real Zeros of Polynomials

Lesson 2.3 Real Zeros of Polynomials. The Division Algorithm. Dividing by a polynomial Set up in long division. 2 terms in divisor (x + 1). How does this go into 1 st two terms in order to eliminate the 1 st term of the dividend. 2x. + 1. Multiply by the divisor

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Lesson 2.3 Real Zeros of Polynomials

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  1. Lesson 2.3Real Zeros of Polynomials

  2. The Division Algorithm

  3. Dividing by a polynomial Set up in long division 2 terms in divisor (x + 1). How does this go into 1st two terms in order to eliminate the 1st term of the dividend. 2x + 1 • Multiply by the divisor • Write product under dividend • Subtract • Carry down next term • Repeat process - 2x2 + 2x - x + 5 - x + 1 - 4 Answer:

  4. HINTS: If a term is missing in the dividend – add a “0” term. If there is a remainder, put it over the divisor and add it to the quotient (answer) Example 1 (x4 – x2 + x) ÷ (x2 - x + 1)

  5. Synthetic Division • Less writing • Uses addition • Setting Up • Divisor must be of the form: x – a • Use only “a” and coefficients of dividend • Write in “zero terms” x – 2: a = 2 x + 3: a = -3 4 5 0 -2 5

  6. 4 5 0 -2 5

  7. Steps • Bring down • Multiply diagonally • Add • Remainder = last addition • Answer • Numbers at bottom are coefficients • Start with 1 degree less than dividend REPEAT

  8. Example 2: (2x3 – 7x2 – 11x – 20) (x – 5)

  9. Example 3: (2x4 – 30x2 – 2x – 1) (x – 4) Problem Set 2.3 (1 – 21 EOO)

  10. The Remainder Theorem If f(x) is divided by x – a , the remainder is r = f(a) The Factor Theorem If f(x) has a factor (x – a) then f(a) = 0

  11. Example 4 Show that (x – 2) and (x + 3) are factors of

  12. Rational Zero Test Every rational zero = Factors of constant term Factors of leading coefficient =

  13. Descartes’ Rule Number of positivereal roots is: ► the number of variations in the signs, or ► less than that by a positive even integer 5x4 – 3x3 + 2x2 – 7x + 1 variations: possible positive real roots:

  14. Example 5 List possible zeros, verify with your calculator which are zeros, and check results with Descartes’ Rule Problems Set 2.3

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