Problem 59 - Elementary and Intermediate Algebra

Question

Factor out \(-1\) from each polynomial. $$ -x^{2}-x+16 $$

Step-by-Step Solution

Answer

The factored polynomial is \(-1(x^{2} + x - 16)\).

Step 1: Identify the Polynomial

The polynomial given is \[-x^{2}-x+16\].We need to factor out \(-1\) from this polynomial.

Step 2: Factor Out -1

To factor out \(-1\), we take out \(-1\) from each term in the polynomial, which is equivalent to multiplying each term by \(-1\).For \(-x^2\), factoring out \(-1\) gives us \(x^2\).For \(-x\), factoring out \(-1\) gives us \(x\).For \(16\), factoring out \(-1\) gives us \(-16\).Thus, the factored form becomes:\[-1(x^{2} + x - 16)\].

Key Concepts

Understanding Algebra Basics

Algebra serves as a foundational component in mathematics, dealing predominantly with symbols and the rules for manipulating those symbols.
In algebra, numbers are often represented by variables like \(x\), \(y\), and other letters. These variables allow us to write expressions and equations in a general form, providing a universal way to talk about mathematical patterns and concepts.
Unlike arithmetic, where we perform operations on specific numbers, algebra allows for operations to be performed on unknown amounts and is used to express real-world problems in a manageable form.

• Variables: Letters that stand in for unknown numbers or values.
• Expressions: Combinations of variables, numbers, and operations.
• Equations: Mathematical statements indicating that two expressions are equal.

Understanding these basics of algebra is crucial for grasping more complex concepts like factoring polynomials.

Exploring Polynomial Expressions

Polynomial expressions are a central concept in algebra.
They are expressions that consist of variables raised to various powers, multiplied by coefficients, and can be represented in the general form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\).
In simpler terms, a polynomial is made up of many terms, where each term includes a constant multiplied by the variable raised to a non-negative integer power.

• Each part (like \(x^2\), \(x\), or constants like 16) is called a "term."
• A polynomial is often organized in order from the highest power to the lowest.
• The coefficients are the numbers in front of the variables and dictate how much of each variable part is included.

Factoring polynomials, as in the provided exercise, is a fundamental operation in algebra that simplifies the expression, making it easier to handle or solve related equations.

Handling Negatives in Polynomials

Working with negatives in polynomials can be tricky, but understanding how to handle them is essential for performing accurate mathematical operations.
When factoring, such as factoring out \(-1\) from a polynomial, each term must be adjusted to account for the negative sign.
This involves changing the sign of each term in the polynomial, as was shown in the original exercise:

• Negative coefficients become positive when \(-1\) is factored out. For example, \(-x^2\) becomes \(x^2\) when factored by \(-1\).
• Positive constant terms become negative after factoring out \(-1\). For instance, 16 becomes \(-16\).

This operation helps simplify the polynomial, revealing a pattern that is often easier to analyze or use in further algebraic operations.