**Problem 8**: Use the Rational Zero Test to find the rational zeros of $ f(x) = x^3 + 3x^2 - x - 3 $.**Explanation**: The Rational Zero Test is a mathematical tool that helps to find possible rational zeros of a polynomial function. It states that any rational zero, in simplest form, $ \frac{p}{q} $, of the polynomial $ f(x) $ is such that $ p $ is a factor of the constant term and $ q $ is a factor of the leading coefficient.For the polynomial $ f(x) = x^3 + 3x^2 - x - 3 $:- The constant term is $-3$.- The leading coefficient is $1$.Factors of $-3$: $\pm 1, \pm 3$Factors of $1$: $\pm 1$Possible rational zeros: $\pm 1, \pm 3$To find the actual rational zeros, substitute these possible values into the polynomial to see which ones evaluate to zero.This test narrows down the search for rational solutions, making it easier to solve polynomial equations.

Here in the given question, we have to find all the real zeroes of the function. fx=x3-3x2-x+3…

Answered: . Use the Rational zero test to find…

Math

Advanced Math

. Use the Rational zero test to find the rational zeros of f(x) = x³ + 3x² – - x – 3

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The main dissimilarity between real and complex roots is that the real roots are obtained as real numbers, in contrast with the complex roots, which are indicated as imaginary numbers. The standard format for depicting any complex number is in the form of a + bi.

Question

**Problem 8**: Use the Rational Zero Test to find the rational zeros of $ f(x) = x^3 + 3x^2 - x - 3 $.

**Explanation**:

The Rational Zero Test is a mathematical tool that helps to find possible rational zeros of a polynomial function. It states that any rational zero, in simplest form, $ \frac{p}{q} $, of the polynomial $ f(x) $ is such that $ p $ is a factor of the constant term and $ q $ is a factor of the leading coefficient.

For the polynomial $ f(x) = x^3 + 3x^2 - x - 3 $:

- The constant term is $-3$.
- The leading coefficient is $1$.

Factors of $-3$: $\pm 1, \pm 3$

Factors of $1$: $\pm 1$

Possible rational zeros: $\pm 1, \pm 3$

To find the actual rational zeros, substitute these possible values into the polynomial to see which ones evaluate to zero.

This test narrows down the search for rational solutions, making it easier to solve polynomial equations.

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