RATIONAL ZERO RATIONAL ZERO THEOREMTHEOREM

Reynaldo B. Pantino, T2Reynaldo B. Pantino, T2

ObjectivesObjectives

To find the rational zeros of any To find the rational zeros of any polynomial functionpolynomial function with the use of; with the use of;

Remainder theoremRemainder theoremFactor theoremFactor theoremSynthetic DivisionSynthetic DivisionFactoringFactoringQuadratic FormulaQuadratic Formula

RecapitulationRecapitulation

Leading CoefficientLeading Coefficient

Constant termConstant term

Remainder theoremRemainder theorem

Synthetic DivisionSynthetic Division

Factor theoremFactor theorem

Unlocking of difficultiesUnlocking of difficulties

What is zero of a function?What is zero of a function?

What is rational zero?What is rational zero?

What is relatively prime integers?What is relatively prime integers?

Remember that;Remember that;

Every rational number can be Every rational number can be written as a quotient of written as a quotient of relatively prime integers.relatively prime integers.

Consider this!Consider this! P(x) = 2xP(x) = 2x33 + 5x + 5x22 – 4x – 3 – 4x – 3

Any rational zero of the function Any rational zero of the function must have a numerator, that is a must have a numerator, that is a factor of -3 (±1 or ±3) and a factor of -3 (±1 or ±3) and a denominator of 2 that is a factor of 2 denominator of 2 that is a factor of 2 (±1 or ±2)(±1 or ±2)an = 2

a0 = – 3

Leading coefficient

Constant term

Let’s define!Let’s define! q p

Illustrative ExamplesIllustrative Examples

Illustrative ExamplesIllustrative ExamplesLet P(x) = xLet P(x) = x33 + 6x + 6x22 + 10x + 3. Find the rational + 10x + 3. Find the rational zeros of P(x). If possible find the zeros.zeros of P(x). If possible find the zeros.

Solution: Solution: (Continuation) (Continuation)

Dividing each p and q, the resulting possibilities Dividing each p and q, the resulting possibilities for p/q are: for p/q are:

±1 ±3or simply

Continuation of the solutionContinuation of the solution::

Try -1 (By synthetic division)Try -1 (By synthetic division)

P(x) = xP(x) = x33 + 6x + 6x22 + 10x + 3 + 10x + 3

Since ;Since ;

P(-1) = -2, -1 is not a zero of P(x).P(-1) = -2, -1 is not a zero of P(x).

1-1-1 6 10 3

1 -1 5

-5 5

-5 -2

Continuation of the solutionContinuation of the solution::

Try -3 (By synthetic division)Try -3 (By synthetic division)

P(x) = xP(x) = x33 + 6x + 6x22 + 10x + 3 + 10x + 3

Since ;Since ;

P(-3) = 0, -3 is a zero of P(x).P(-3) = 0, -3 is a zero of P(x).

1-3-3 6 10 3

1 -3 3

-9 1

-3 0

Continuation of the solutionContinuation of the solution::Try -3 (By synthetic division)Try -3 (By synthetic division)

P(x) = xP(x) = x33 + 6x + 6x22 + 10x + 3 + 10x + 3

The depressed equation is xThe depressed equation is x22 + 3x + 1 = 0. + 3x + 1 = 0.

Using the results of the synthetic division above;Using the results of the synthetic division above;

P(x) = (x + 3)((xP(x) = (x + 3)((x22 + 3x + 1). + 3x + 1).

1-3-3 6 10 3

1 -3 3

-9 1

-3 0

Continuation of the solutionContinuation of the solution::Since the equation xSince the equation x22 + 3x + 1 = 0 is quadratic, + 3x + 1 = 0 is quadratic, use the quadratic formula to find the other zeros use the quadratic formula to find the other zeros of P(x). of P(x).

The Quadratic Formula:The Quadratic Formula:

Continuation of the solutionContinuation of the solution::Since the equation xSince the equation x22 + 3x + 1 = 0 is quadratic, + 3x + 1 = 0 is quadratic, use the quadratic formula to find the other zeros use the quadratic formula to find the other zeros of P(x). of P(x).

Continuation of the solutionContinuation of the solution::Since the equation xSince the equation x22 + 3x + 1 = 0 is quadratic, + 3x + 1 = 0 is quadratic, use the quadratic formula to find the other zeros use the quadratic formula to find the other zeros of P(x). of P(x).

Test YourselfTest Yourself::Find the rational zeros of the polynomial Find the rational zeros of the polynomial function if they exist. If possible, find the other function if they exist. If possible, find the other zeros. Then write the function in factored form.zeros. Then write the function in factored form.

1.) P(x) = x3 – x2 – 3x + 3

2.) P(x) = x3 – 4x2 + 2x + 4

3.) P(x) = x3 – x2 – 4x – 2

4.) P(x) = x3 – 2x2 + x + 4

5.) P(x) = 2x3 – 3x2 – 7x + 6

ASSIGNMENTSASSIGNMENTS

Answers numbers 6 to 12 on Answers numbers 6 to 12 on page 110, page 110,

(Advanced Algebra, (Advanced Algebra, Trigonometry and Statistics) Trigonometry and Statistics)