2.6 – Find Rational zeros

1. 2.6 – Find Rational zeros Coach Bianco

2. 2.6 – Find Rational zeros • Georgia Performance Standards: • MM3A3a – Find real and complex roots of higher degree polynomial equations using the factor theorem, remainder theorem, rational root theorem, and fundamental theorem of algebra, incorporating complex and radical conjugates. • MM3A3d – Solve a variety of types of equations by appropriate means choosing among mental calculation, pencil and paper, or appropriate technology.

3. Find zeros when the leading coefficient is 1 • Find all real zeros of f(x) = x3 – 7x2 + 14x – 8. • *The rational root theorem states if anxn + … + a1x + a0 has integer coefficients, then every rational zero of f has the form =

4. Find zeros when the leading coefficient is 1 • Find all real zeros of f(x) = x3 – 7x2 + 14x – 8 • Steps: • List the possible rational zeros. The leading coefficient is 1 and the constant term is -8. The possible real zeros are: x = +- 1, +-2, +-4, +- 8 • Test these zeros using synthetic division. Test x = 4: 4 -7 14 -8 4 -12 8 1 -3 2 0 *Because 4 is a zero off(x) = (x -4)(x2 – 3x + 2). 4 is a zero

5. Find zeros when the leading coefficient is 1 • Find all real zeros of f(x) = x3 – 7x2 + 14x – 8 • Steps: • List the possible rational zeros. The leading coefficient is 1 and the constant term is -8. The possible real zeros are: x = +- 1, +-2, +-4, +- 8 • Test these zeros using synthetic division. Test x = 4: • Factor the trinomial and use the factor theorem. • f(x) = (x -4)(x2 – 3x + 2) = (x – 4)(x – 2)(x – 1) • The zeros are 1, 2, and 4.

6. Find zeros when the leading coefficient is not 1 • Find all real zeros of f(x) = 3x3 – 17x2 + 18x + 8. • Steps: • List the possible rational zeros • Choose a reasonable value to check using the graph of the function • Check with synthetic division • Factor out a binomial using the result of synthetic division

7. Find zeros when the leading coefficient is not 1 • Find all real zeros of f(x) = 3x3 – 17x2 + 18x + 8. • Possible zeros: +- 1, 2, 4, 8, - 3 -17 18 8 -1 6 -8 3 -18 24 0 - is a zero

8. Find zeros when the leading coefficient is not 1 • Find all real zeros of f(x) = 3x3 – 17x2 + 18x + 8. • Steps: • List the possible rational zeros • Choose a reasonable value to check using the graph of the function • Check with synthetic division • Factor out a binomial using the result of synthetic division

9. Find zeros when the leading coefficient is not 1 • Find all real zeros of f(x) = 3x3 – 17x2 + 18x + 8. • *The real zeros are , 2, and 4. -

10. Guided practice: Page 90 (1-6)