Question

Determine whether the binomial is a factor of the polynomial function. $$ f(x)=2 x^3+5 x^2-37 x-60 ; x-4 $$

Short answer

Yes, the binomial \(x-4\) is a factor of the polynomial function \(f(x)=2 x^3+5 x^2-37 x-60\).

Step-by-step solution

Identify the root of the binomial.

The root of a binomial \(x-a\) is \(x=a\), so it follows that the root for \(x-4\) is \(x=4\). Hence, \(a=4\).

Substitute the root value in the polynomial function.

Substitute the root value \(x=4\) into the polynomial function \(f(x)=2x^3 + 5x^2 - 37x - 60\). This gives: \(f(4)=2(4)^3 + 5(4)^2 - 37(4) - 60\).

Calculate the value of the function at the given root.

Calculate the value of \(f(4)\). Simplifying the equation obtained previously results in: \(f(4)=2(64) + 5(16) - 37(4) - 60 = 128 + 80 - 148 - 60 = 0\).

Analyze the result

Since the value of the polynomial function at the root of the binomial \(x=4\) is 0, it indicates that \(x-4\) is indeed a factor of \(f(x)\).

Key concepts

Factorization

Factorization is the process of breaking down a larger mathematical expression into a product of simpler factors. It helps in simplifying expressions and solving equations more easily. In the context of polynomials, factorization means writing the polynomial as a product of its simpler polynomial factors.

• For example, in the given exercise, the polynomial function is expressed as \( f(x) = 2x^3 + 5x^2 - 37x - 60 \).
• Our task is to determine if the binomial \( x - 4 \) is a factor of this polynomial.
• When \( x - 4 \) is a factor, the polynomial can be divided by this binomial without leaving a remainder.

To check for factors, we can use either synthetic division or substitute the root into the polynomial, as shown in the given step-by-step solution. If the result is zero, then the binomial is a factor of the polynomial.

Polynomial Roots

Polynomial roots are the values of \( x \) for which the polynomial equals zero. They are also known as the solutions or zeros of the polynomial. Identifying roots is crucial because they tell us where a function crosses or touches the x-axis on a graph.

• In the exercise, the expression \( x - 4 \) leads us to a root, \( x = 4 \).
• By substituting \( x = 4 \) into the polynomial function and finding that the function evaluates to zero, we confirm a root at \( x = 4 \).
• This root indicates that the polynomial can be factored with \( x - 4 \) as one of its factors.

Polynomial roots can be real or complex, and their multiplicity (how many times they appear) also plays a significant role.

Binomial Theorem

The Binomial Theorem provides a way to expand expressions raised to a power, specifically for binomials. While the Binomial Theorem itself was not directly applied in the solution, it provides a foundational understanding of how binomials behave within complex expressions.

• A binomial is an algebraic expression with two terms, such as \( x - 4 \).
• In our exercise, evaluating whether \( x - 4 \) is a factor involves understanding the root of the binomial and its relationship with the polynomial.
• The theorem is often symbolized by the expression:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]

Understanding binomials' properties and behaviors helps in manipulating and factoring polynomials efficiently. Even if the theorem itself is not directly used, the fundamental principles underpin many algebraic techniques involved in polynomial division and factorization.