**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 Consider the polynomial function fx=x3-3x2-x+3 To find all possible zero of the polynomial need to apply the Rational Root Test to the polynomial. According to the Rational Root Test: A polynomial of the form: fx=anxn+an-1xn-1+…+a2x2+a1x+a0where an,an-1,…a0 are integer Then an is the leading coefficient and a0 constant term but an≠0 and a0≠0 . Then every rational root is of the form: pq=±factors of a0 ±factors of an Here in the given polynomial. 1 is the leading coefficient and 3 is the constant term. So, the possible rational zeroes can be calculated as: pq=±factors of a0 ±factors of an=±factors of 3 ±factors of 1=±1,±3 If a polynomial can be divided by its linear factor with no remainder, then that zero of that linear factor will be the root of the polynomial. Put the value x = 1 in the given polynomial. fx=x3-3x2-x+3At x=1:f1=13-312-1+3=1-3-1+3=0f1=0⇒x=1 is the root of the given function.at x=-1:f-1=-13-3-12--1+3=-1-3+1+3=0f-1=0⇒x=-1 is the root of the given function.At x=3f3=33-332-3+3=27-27-3+3=0f3=0⇒x=3 is the root of the given function.By linear factorization theorem, the polynomial will have the same number of factors as its degree and the polynomial can be written as: fx=x3-3x2-x+3fx=x-3x+1x-1The solutions are x=1, x=-1, x=3 Thus, the rational zeros of the given function are x=-1, x=1 and x=3.

Answered: . Use the Rational zero test to find…

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. 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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Roots

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.

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**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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