Question

Apply the distributive property. $$-x(x-6)$$

Short answer

-x^2 + 6x

Step-by-step solution

Identify the parts

Identify the values that will be used in the distributive property. In this case, the entire term -x will be distributed over the terms in the parentheses, setting a equal to -x, b equal to x, and c equal to -6.

Apply the distributive property

Apply the distributive property to the given expression. Distribute -x across the terms in the parentheses to get the expression as \(-x \cdot x + (-x) \cdot -6\).

Simplify the expression

Simplify the expression by multiplying the values together to obtain the final result, which simplifies to \(-x^2 + 6x\).

Key concepts

Algebraic Expressions

In algebra, an algebraic expression is a combination of numbers, variables (like x), and operations (such as addition, subtraction, multiplication, and division) that represents a particular value or set of values. Understanding algebraic expressions is crucial for solving many types of math problems.

For instance, consider the algebraic expression \( -x(x-6) \). Here, x is our variable, and the expression includes multiplication and subtraction. The beauty of algebra is that it allows us to work with unknowns and find their possible values by performing operations similar to those we do with actual numbers.

To handle algebraic expressions correctly, it's essential to understand the standard order of operations: first inside the parentheses, then exponents, followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right). This hierarchy ensures that we simplify expressions systematically and reach the correct solution.

Simplifying Expressions

The art of simplifying expressions involves reducing them to their most basic form without changing their value. Simplification can involve combining like terms, factoring, expanding expressions, and applying mathematical properties such as the distributive property.

In our exercise, simplification is accomplished by applying the distributive property. We simplify \( -x(x-6) \) by distributing \( -x \) across the terms inside the parentheses. This step transforms the expression into the simplified form \(-x^2 + 6x\).

Key to simplification is always to remember to perform multiplication or division before addition or subtraction, unless grouping symbols like parentheses indicate otherwise. Simplifying expressions is not just about making them shorter; it's about making them clearer and easier to work with, especially when solving equations or evaluating expressions for given values of the variables.

Multiplication of Polynomials

When it comes to the multiplication of polynomials, it's like performing a dance with numbers and variables—we follow specific steps to ensure the move is completed without a misstep. A polynomial is an algebraic expression containing one or several terms, each including a variable raised to a whole number exponent.

The distributive property is a guiding star for polynomial multiplication. It allows us to multiply each term of one polynomial by each term of another. For example, taking \( -x \) and multiplying it by each term in \(x-6\) is actually an instance of polynomial multiplication.

By distributing \( -x \) across \(x-6\), we're multiplying two polynomials: \( -x \) and \( x-6 \). The result, \( -x^{2} + 6x \), is also a polynomial, but it's been simplified from the original multiplication form to its expanded form. This process is fundamental in algebra and essential for understanding higher-level mathematics, including calculus and beyond.