Problem 10 - Algebra 2

Question

Determine whether each binomial is a factor of \(x^{3}+4 x^{2}+x-6\) $$ x+2 $$

Step-by-Step Solution

Answer

\(x + 2\) is a factor of \(x^{3}+4 x^{2}+x-6\) because substituting \(x = -2\) into the polynomial results in zero.

Step 1: Apply the Factor Theorem

To determine if a binomial \(x + a\) is a factor of a polynomial, we apply the Factor Theorem. According to the theorem, \(x + a\) is a factor if and only if the polynomial equals zero when \(x = -a\). Hence, we will substitute \(x = -2\) into the polynomial and see if it equals zero.

Step 2: Substitute the value into the polynomial

Replace \(x\) with \( -2\) in the polynomial \(x^{3}+4x^{2}+x-6\) and simplify. This gives us\((-2)^{3} + 4(-2)^{2} + (-2) - 6\).

Step 3: Simplify the expression

Simplify the expression by performing the operations: \( (-2)^{3} = -8\), \(( -2)^{2} = 4\), thus giving us \( -8 + 4(4) - 2 - 6 = -8 + 16 - 2 - 6\). When we further simplify, we obtain \( -8 + 16 - 2 - 6 = 0\).

Step 4: State the conclusion

Since substituting \(x = -2\) into the polynomial yields zero, according to the Factor Theorem, \(x + 2\) is indeed a factor of the polynomial \(x^{3}+4x^{2}+x-6\).

Key Concepts

Binomial Factors

Understanding binomial factors is crucial when learning about polynomials and their properties. Binomial factors are expressions containing two terms, typically in the form of \(x + a\) or \(x - a\), where \(a\) is a constant. When a polynomial can be divided by a binomial without leaving a remainder, that binomial is considered a factor of the polynomial.

In the case of the exercise, \(x + 2\) is tested as a potential factor of the given polynomial \(x^{3} + 4x^{2} + x - 6\). To verify if it is indeed a factor, the Factor Theorem is applied. According to this theorem, \(x + 2\) is a factor if and only if the polynomial evaluates to zero when \(x = -2\). This is a powerful tool because it simplifies the process of factoring polynomials, avoiding more time-consuming methods such as long division or synthetic division if only checking for a single binomial factor.

In summary, recognizing and testing binomial factors form an integral part of polynomial factoring and are essential skills for algebra students.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial to its simplest form by performing arithmetic operations and combining like terms. It is a process fundamental to algebra and helps in solving equations, graphing functions, and simplifying complex expressions.

In relation to the exercise provided, after substituting \(x = -2\) into the polynomial \(x^{3} + 4x^{2} + x - 6\), the expression \( (-2)^{3} + 4(-2)^{2} + (-2) - 6\) must be simplified. To do this, one must carefully follow the order of operations: address exponents first, then multiplication and division, followed by addition and subtraction.

Furthermore, simplification allows for the clearer identification of patterns, such as the potential for binomial factors, as seen when the expression reduces to zero, confirming \(x + 2\) as a factor. The skill of polynomial simplification is not only crucial for this exercise but also for a wide array of algebraic applications.

Zeroes of Polynomial

Zeroes of a polynomial, also known as roots or solutions, are the values of \(x\) for which the polynomial equals zero. These values represent the intersection points of the polynomial function with the \(x\)-axis when graphed. Identifying the zeroes of polynomials is important in understanding the behavior of the polynomial function and in solving polynomial equations for various applications.

In the context of our exercise, by applying the Factor Theorem, we find that when \(x = -2\), the polynomial \(x^{3} + 4x^{2} + x - 6\) equals zero. Thus, \(x = -2\) is one of the zeroes of the polynomial. Since \(x + 2\) yields zero when substituted into the polynomial, it reinforces the fact that \(x + 2\) is a binomial factor of the polynomial.

In other words, every time we identify a zero of a polynomial, we have also found a factor, and vice versa. Consequently, the concept of zeroes is intricately linked to the factors of a polynomial, and understanding one aids in understanding the other. This concept is vital in fields ranging from algebra to calculus and beyond.