36f(x) = X^3 - 4x^2 + 9x - 36 - Wyzant

Subject ZIP Search Search Find an Online Tutor Now Ask Ask a Question For Free Login Rational Therom f(x) = x^3 - 4x^2 + 9x - 36f(x) = x^3 - 4x^2 + 9x - 36

Use the Rational therom to fibd the real zeros.

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Rational Zero Theorem provides a complete list of possible rational roots of the polynomial equation anxn + an-1xn-1 + .... + a2x2 + a1x + a0 = 0 , where all coefficients are integers. Possible zeros are p/q , were p is from set of factors for a0 , q is from set of factors of an .~~~~~~~~~~~~~x3 - 4x2 + 9x - 36 = 0 ----> a3 = 1 ------> q is from {±1} , a0 = -36 ------> p is from {±1,±2, ±3, ±4,±6,±9,±12,±18,±36} p/q = ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36 , if we plug in each number , one by one, we can see that x = 4 is a zero of given function and that will be the only zero in the set of real numbers. ƒ(x) = (x - 4)(x2 + 9)

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First, note that there are 3 sign changes. That means that there are 3 or 1 possible positive real roots and since f(-1) has no sigh changes, there are no negative real roots.

To find the possible rational roots, find all factors of the last term (36) and all factors of the leading coefficient (1) and divide the factors of 36 by the factors of 1.

This means that the roots will come out of the set {1,2,3,4,6,9,12,18,36}, all positive.

Try 1 first by finding f(1). Since this doesn't equal 0, try using synthetic division dividing by 2. If you continue through the set, you will find that 4 will work. You will end up with a quadratic equation which can be solved by other means.

I hope this helps.

Kevin

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If a polynomial function, written in descending order of the exponents, has integer coefficients, then any rational zero must be of the form +- p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

The constant term is 36

The leading coefficient is 4

The p factors are 2,3,4,9,12,18

The q factor is 1

Al the possible values of +-p/q are +-2/1, +-3/1, +-4/1, +-6/1, +-9/1, +-12/1, +-18/1

Simplified Values of +-p/q are +-2, +-3, +-4, +-6, +-9, +-12, +-18

Trying each value shows that 4 is the only value the works

43 – (4)(42) + (9)(4) -36 = 0

64 – 64 +36 – 36 = 0

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