Question

Determine whether the binomial is a factor of the polynomial function. $$ g(x)=8 x^5-58 x^4+60 x^3+140 ; x-6 $$

Short answer

No, the binomial \(x-6\) is not a factor of the polynomial \(g(x)=8 x^5-58 x^4+60 x^3+140\) since substituting \(x=6\) into the polynomial results in a value of \(55552\), and not zero.

Step-by-step solution

Isolate the constant in the binomial

The binomial is \(x-6\), so \(a=6\). This is the value that will be substituted into the polynomial.

Substitute the constant into the polynomial

Substitute \(x=6\) into the polynomial function \(g(x)=8 x^5-58 x^4+60 x^3+140\). This gives \( g(6) = 8(6^5) - 58(6^4) + 60(6^3) + 140\).

Calculate the value of the polynomial at x=6

Compute the value of \( g(6) = 8(7776) - 58(1296) + 60(216) + 140 = 55552 \).

Determining if the binomial is a factor

To check if the binomial \(x-6\) is a factor of the polynomial, the calculated value for \(g(6)\) is compared to zero. Since \(55552 \neq 0\), the binomial \(x-6\) is not a factor of the given polynomial.

Key concepts

Binomial Factors

A binomial factor in the context of polynomials is a two-term expression, such as \( x - a \), that might evenly divide a polynomial function without leaving a remainder. Understanding if a binomial is a factor of a polynomial is significant because it allows us to break down more complex equations into simpler components, which can be useful in various mathematical analyses, including solving for roots and simplifying expressions.

To determine if a binomial is a factor, we follow a process similar to the original exercise. The particular binomial \( x - a \) implies that \( a \) is a root of the polynomial if the binomial is indeed a factor. For the polynomial \( g(x) \) in the original problem, the binomial in question was \( x - 6 \) where \( 6 \) is the value being tested. By substituting \( x = 6 \) into the polynomial and evaluating it, if the result were 0, \( x - 6 \) would be a factor of \( g(x) \). However, since the evaluation yielded \( 55552 \), which is not zero, it's clear that \( x - 6 \) is not a factor of the polynomial function.

Factor Theorem

The Factor Theorem is a fundamental principle connecting polynomial functions to their roots. It states that a binomial factor \( x - a \) is indeed a factor of a polynomial \( f(x) \) if and only if the polynomial evaluates to zero when \( a \) is substituted in for \( x \), meaning \( f(a) = 0 \).

If we take the example provided in the exercise, the Factor Theorem was applied when testing whether \( x - 6 \) is a factor of \( g(x) \). We substituted the value \( a = 6 \) into the polynomial to see if it satisfies the condition of the Factor Theorem. Since \( g(6) \) did not equal zero, the theorem tells us that \( x - 6 \) is not a factor. This theorem is a powerful tool for solving polynomials because once a factor is identified, it can be used to factorize the polynomial into smaller parts, often leading to a discovery of all zeroes of the polynomial.

Polynomial Functions

Polynomial functions are algebraic expressions consisting of variables and coefficients, structured in whole number exponents on the variables. They're written in the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \) where \( n \) is a non-negative integer, and \( a_n \) (for \( n = 0, 1, 2, ... \) are constants with \( a_n ≠ 0 \).

A key characteristic of polynomial functions is that they are continuous and smooth, meaning they have no breaks, holes, or sharp corners on their graphs. The given function \( g(x) \) in the exercise is a polynomial of degree five, which means its graph can potentially cross the x-axis up to five times. Understanding the structure of polynomial functions can greatly aid in sketching their graphs and solving for their roots. Additionally, various theorems and properties, such as the Factor Theorem and the idea of binomial factors, allow for deeper analysis and simplification of these functions for easier calculation and understanding of their behavior.