Question

Determine whether the binomial is a factor of the polynomial function. $$ g(x)=3 x^3-28 x^2+29 x+140 ; x+7 $$

Short answer

No, the binomial \(x + 7\) is not a factor of the polynomial function \(g(x) = 3x^3 - 28x^2 + 29x + 140\), since the value of g(-7) is not equal to zero.

Step-by-step solution

Apply Remainder Theorem

The Remainder Theorem states that if \(f(a) = 0\) for a polynomial \(f(x)\), then \(x - a\) is a factor of \(f(x)\). In this case, to check if \(x + 7\) is a factor of \(g(x)\), check if \(g(-7) = 0\).

Substitute \(x\) in the given polynomial

Substitute \(-7\) for \(x\) in the polynomial \(g(x) = 3x^3 - 28x^2 + 29x + 140\). It becomes \(g(-7) = 3(-7)^3 - 28(-7)^2 + 29(-7) + 140\).

Calculate the Result

Carry out the calculations to find the value of \(g(-7)\). Calculate the cube, square and product terms individually first, that is, \(3(-7)^3 = -1029\), \(-28(-7)^2 = -1372\), \(29(-7) = -203\) and then add all these values along with 140. After calculation, we find that \(g(-7) = -2464 \neq 0\).

Key concepts

Remainder Theorem

The Remainder Theorem is a fundamental concept in algebra that helps determine whether a binomial is a factor of a polynomial. It's based on a simple idea: if the polynomial function, say \(f(x)\), is divided by a binomial \(x - a\) and it results in a remainder of zero, this means that \(x - a\) is a factor. To use the Remainder Theorem, substitute the value \(a\) into the polynomial function and calculate \(f(a)\).

If \(f(a) = 0\), then \(x - a\) is a factor. For instance, given a polynomial \(g(x)\), if we need to check if \(x + 7\) is a factor, we substitute \(-7\) for \(x\) (since the binomial is in the form \(x - a = x - (-7)\). Calculate \(g(-7)\) and verify if the result equals zero. This process provides a quick and efficient method to test factorization without performing complex polynomial division.

Binomials

Binomials are algebraic expressions that consist of two terms connected by a plus or minus sign. They are among the simplest forms of polynomials and are usually expressed as \(ax + b\).

In our exercise, the binomial is \(x + 7\). Binomials are essential components in polynomial operations because they can act as potential factors to larger polynomial expressions. Checking if a binomial is a factor of a polynomial involves substituting its root (which is the inverse of the constant term) into the polynomial using the Remainder Theorem.

Understanding binomials can also help build a robust foundation for learning other algebraic concepts such as factoring, simplifying expressions, and solving equations. They serve as the building blocks for polynomial equations used in many branches of mathematics.

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables and coefficients. They are in the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\). These functions can describe a wide range of natural phenomena and data patterns. The highest power of the variable is called the degree of the polynomial, and it dictates the polynomial's most significant behavior.

In our exercise, the polynomial function \(g(x) = 3x^3 - 28x^2 + 29x + 140\) is a cubic polynomial, meaning its highest degree is 3.

Each term of a polynomial contains a coefficient, such as \(3, -28, 29,\) and a constant term \(140\). These coefficients influence the shape and direction of the polynomial graph. Polynomial functions are versatile, used for interpolating data points and modeling curves in physics, biology, economics, and other fields. A clear understanding of these functions and their properties is vital for tackling more complex mathematical problems.