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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.

FIND ZEROS WHEN THE LEADING COEFFICIENT IS 1Find 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

=

FIND ZEROS WHEN THE LEADING COEFFICIENT IS 1Find all real zeros of f(x) = x3 – 7x2 + 14x – 8

Steps:

1. 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

2. Test these zeros using synthetic division. Test x = 4:1 -7 14 -8

4 -12 8

1 -3 2 0

4 is a zero

4 *Because 4 is a zero of f(x) = (x -4)(x2 – 3x + 2).

FIND ZEROS WHEN THE LEADING COEFFICIENT IS 1Find all real zeros of f(x) = x3 – 7x2 + 14x – 8

Steps:

1. 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

2. Test these zeros using synthetic division. Test x = 4:

3. 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.

FIND ZEROS WHEN THE LEADING COEFFICIENT IS NOT 1

Find all real zeros of f(x) = 3x3 – 17x2 + 18x + 8.

Steps:

1. List the possible rational zeros

2. Choose a reasonable value to check using the graph of the function

3. Check with synthetic division

4. Factor out a binomial using the result of synthetic division

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

-

FIND ZEROS WHEN THE LEADING COEFFICIENT IS NOT 1

Find all real zeros of f(x) = 3x3 – 17x2 + 18x + 8.

Steps:

1. List the possible rational zeros

2. Choose a reasonable value to check using the graph of the function

3. Check with synthetic division

4. Factor out a binomial using the result of synthetic division

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. -

GUIDED PRACTICE: PAGE 90 (1-6)