Question

USING THE DISTRIBUTIVE PROPERTY Use the distributive property to simplify the expression. $$(4+x)(-6 x)$$

Short answer

The simplified form of the expression \( (4+x) \cdot (-6x) \) is -24x - 6x².

Step-by-step solution

Apply the distributive property

Applying the distributive property to the expression \( (4+x) \cdot (-6x) \) means that every term inside the parentheses must be multiplied by the factor outside the parentheses. So, we have to multiply -6x by each term inside the parentheses (4 and x) separately.

Multiply -6x by each term inside the parentheses

Multiply -6x by 4 and -6x by x. When -6x is multiplied by 4, the result is -24x. When -6x is multiplied by x, the result is -6x².

Simplify the multiplication results into one expression

Combine the multiplication results from Step 2 into one simplified expression. So, the simplified expression is -24x - 6x².

Key concepts

Simplifying algebraic expressions

Simplifying algebraic expressions is a vital skill in algebra that involves reducing an expression to its simplest form. This makes the expression easier to work with and understand. The process typically includes various methods such as the application of the distributive property, which was showcased in our exercise example with the expression \( (4+x) \cdot (-6x) \).

To simplify, we first need to ensure we multiply each term inside the parentheses by the outside term, respecting the operations involved. After applying the distributive property, we might need to further simplify the expression by combining like terms or by performing any additional arithmetic to find a more concise version of the original expression.

Multiplication in algebra

When we talk about multiplication in algebra, we refer to the process of combining algebraic terms through multiplication. In contrast to basic arithmetic multiplication, where we deal with numbers alone, algebra introduces variables (like x in our example) into the mix. This inclusion of variables can make multiplication seem more complex at first.

Let's take the step in our exercise where we multiply -6x by 4 to get -24x. We're combining a constant, 4, with a variable term, -6x. To multiply them, we multiply the coefficients (in this case, -6 and 4) and keep the variable part (x) the same. But if we also multiply -6x by another variable term, for instance, x, we use the exponent rules and obtain -6x², indicating that x has been squared. This operation showcases one of the fundamental aspects of multiplication in algebra — combining constants, variables, and their powers to create new algebraic expressions.

Combining like terms

Combining like terms is a technique used to further simplify an algebraic expression. 'Like terms' are terms that have the same variables raised to the same power. For instance, 3x and 5x are like terms because they both have the single variable 'x' to the power of 1. However, 3x and 3x² are not like terms since the powers of 'x' are different.

Returning to our original problem, once we applied the distributive property, we obtained the terms -24x and -6x². These two are not like terms because the variable x is raised to different powers in each term. Hence, they cannot be combined further. But if we had terms like 4x and -24x, they could be combined to simplify the expression, which would result in -20x since we add the coefficients and keep the variable the same. Recognizing and combining like terms is essential for simplifying algebraic expressions effectively.