Question

Use the Distributive Property to remove the parentheses. $$ 4 x(x+3) $$

Short answer

The expression simplifies to \[4x^2 + 12x\].

Step-by-step solution

Identify the terms

The given expression is \[4x(x+3)\]. Identify the coefficient outside the parentheses and the terms inside the parentheses.

Apply the Distributive Property

According to the Distributive Property, \[a(b+c) = ab + ac\]. In this case, let \(a = 4x\), \(b = x\), and \(c = 3\). Distribute \(4x\) to each term inside the parentheses.

Multiply the terms

First, multiply \(4x\) with \(x\): \[4x \times x = 4x^2\]. Next, multiply \(4x\) with \(3\): \[4x \times 3 = 12x\].

Combine the results

Combine the results from the previous step to get the final expression: \[4x^2 + 12x\].

Key concepts

algebraic expressions

An algebraic expression is a mathematical phrase that includes numbers, variables (like x or y), and arithmetic operations (such as addition, subtraction, multiplication, and division). They don't have an equal sign, which distinguishes them from equations.

In the given exercise, \(4x(x+3)\) is an algebraic expression. It consists of:

• A coefficient (4)
• A variable term (x)
• An operation (multiplication and addition)
Algrebraic expressions could be simplified using different properties of numbers, such as the Distributive Property.
Simplifying algebraic expressions helps us to better understand their values and relationships. In this case, simplifying \(4x(x+3)\) using the Distributive Property makes it easier to work with or solve when used in equations or functions.

polynomials

A polynomial is a special kind of algebraic expression where the terms are only added, subtracted, or multiplied together. Polynomials consist of terms that are non-negative integer powers of the variable in the expression.

• For example, \(4x^2 + 12x\) is a polynomial with two terms.

The highest power of x in a polynomial is called the degree of the polynomial. In this case, the degree of \(4x^2 + 12x\) is 2, because the highest power of x is 2.

Polynomials are not just limited to single variables; they can have more than one variable. For example, \(3x^2 + 2xy + y^2\) is a polynomial in two variables, x and y.

In the given exercise, when we distribute \(4x\) using the Distributive Property, we are simplifying the expression into a polynomial form: \(4x^2 + 12x\). This makes it easier to work with in further mathematical operations or solving equations.

multiplication

Multiplication is a mathematical operation where you combine equal groups. In algebra, it often involves numbers and variables. In the given exercise, we multiply terms by using the Distributive Property.

• The Distributive Property states that \(a(b+c) = ab + ac\).

In this case, after identifying \(a = 4x\), \(b = x\), and \(c = 3\), we distribute \(4x\) to each term inside the parentheses.
• First, we multiply \(4x\) with \(x : 4x \times x = 4x^2\).
• Second, we multiply \(4x\) with \(3 : 4x \times 3 = 12x\).

Combining these results gives us the simplified polynomial \(4x^2 + 12x\). Through these steps, you can see how multiplication is used to break down and simplify algebraic expressions using the Distributive Property.