Factor the polynomial function f(x) . Then solve the equation f(x)=0. f(x)=x^3-x^2-14 x+24

Name: Problem 43, Chapter 4: Algebra and Trigonometry
Uploaded: 2020-07-01T21:20:33-08:00
Duration: 1 min 33 s
Channel: Ankit Gupta
Description: Factor the polynomial function f(x) . Then solve the equation f(x)=0. f(x)=x^3-x^2-14 x+24

step 1 FX is equal to x -cube minus x square minus 14x plus 24. Let's try x minus 2 as a linear factor

Factor the polynomial function f(x) . Then solve the...

Key Concepts

Zero Product Property

The Zero Product Property is a basic principle stating that if the product of several factors equals zero, then at least one of the factors must equal zero. This property is applied after factoring a polynomial, as it allows one to set each factor equal to zero and solve for the variable, thus finding all the roots of the original equation.

Polynomial Division

Polynomial division, including synthetic division or long division, is used to divide a polynomial by one of its factors. When a factor is identified using the Factor Theorem or Rational Root Theorem, dividing the polynomial by that factor helps to reduce the degree of the polynomial and identify additional factors or roots.

Rational Root Theorem

The Rational Root Theorem provides a systematic method to list all possible rational roots of a polynomial equation. It states that any possible rational root, written as a fraction p/q in simplest form, must have a numerator p that divides the constant term and a denominator q that divides the leading coefficient. This helps narrow down the candidates for roots that can be tested.

Polynomial Factoring

Polynomial factoring is the process of rewriting a polynomial as a product of two or more simpler polynomials. This technique is essential for simplifying expressions and solving equations because it can reveal the roots or zeros of the polynomial when it is expressed as a product of its factors.

Factor Theorem

The Factor Theorem is a specific case of the remainder theorem, stating that if a polynomial f(x) has a value of zero at x = a, then (x - a) is a factor of f(x). This theorem is a fundamental tool in polynomial factorization because identifying even one root simplifies the polynomial and makes further factorization easier.

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