Find only the rational zeros of the function. f(x)=x^4-3 x^3-9 x^2-3 x-10

Name: Problem 74, Chapter 4: Algebra and Trigonometry
Uploaded: 2020-07-01T18:36:22-08:00
Duration: 2 min 58 s
Channel: Ankit Gupta
Description: Find only the rational zeros of the function. f(x)=x^4-3 x^3-9 x^2-3 x-10

step 1 We have given the polynomial y equals to fx equals to x to the power 4 minus 3x cube minus 9 x s

Find only the rational zeros of the function. f(x)=x^4-3...

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a tool used to identify all possible rational candidates for the roots of a polynomial equation with integer coefficients. It states that any rational solution, expressed as a fraction in lowest terms, must have a numerator that divides the constant term and a denominator that divides the leading coefficient. This theorem helps narrow down the list of potential rational zeros that need to be tested, making the problem of finding exact roots more manageable.

Factor Theorem

The Factor Theorem is a principle that connects the roots of a polynomial to its factors. It states that if a polynomial f(x) has a zero at x = r, then (x - r) is a factor of the polynomial. This theorem is crucial in the process of polynomial factorization because once a candidate root is verified (typically using the Rational Root Theorem), the polynomial can be divided by (x - r) to reduce its degree and simplify further analysis.

Synthetic Division

Synthetic division is a streamlined method of dividing a polynomial by a binomial of the form (x - r), where r is a candidate zero. It simplifies the division process compared to traditional long division by focusing only on the coefficients of the polynomial. Synthetic division is especially useful when testing each possible rational root obtained from the Rational Root Theorem, as it quickly determines whether a candidate is indeed a zero of the polynomial.

Polynomial Factorization

Polynomial factorization involves breaking down a complex polynomial into a product of simpler polynomials or factors, which can include linear or irreducible quadratic factors. Factorization is a fundamental technique for solving polynomial equations because it allows one to set each factor equal to zero in order to find the zeros of the original polynomial. This method is often used after identifying at least one root using the Rational Root Theorem and synthetic division.

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