Question

Remove the brackets from the given expression: \((x+1)(x-3) x\)

Short answer

Question: Remove the brackets from the expression \(x(x+1)(x-3)\). Answer: The expression with the brackets removed is \(x^3 - 2x^2 - 3x\).

Step-by-step solution

Multiply the two binomials

Let's first multiply the two binomials \((x+1)(x-3)\) using the distributive property (also known as the FOIL method). The distributive property states that \(a(b+c) = ab + ac\). In this case, let \(a = x\), \(b = x + 1\), and \(c = x - 3\). Using the distributive property to multiply the terms inside the brackets, we get: \((x+1)(x-3) = x(x-3) + 1(x-3)\) Now, distribute each term: \(x(x-3) = x^2 - 3x\) \(1(x-3) = x - 3\) Combine the two expressions: \(x^2 - 3x + x - 3\)

Simplify the expression

Now, we need to simplify the expression obtained in Step 1 by combining like terms. In this case, we have two x terms, which we can add together: \(-3x + x = (-3 + 1)x = -2x\) So our simplified expression is: \(x^2 - 2x - 3\)

Distribute x to the simplified expression

Finally, we need to distribute the x term outside the brackets: \(x(x^2 - 2x - 3)\) Using the distributive property, we get: \(x(x^2) - x(2x) - x(3)\) Which simplifies to: \(x^3 - 2x^2 - 3x\) So, the expression with the brackets removed is: \(\boxed{x^3 - 2x^2 - 3x}\)

Key concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that helps us simplify expressions involving multiplication over addition or subtraction. When we say the distributive property, we often refer to the equation: \( a(b + c) = ab + ac \). Here, we distribute the term \( a \) to both \( b \) and \( c \), multiplying \( a \) with each.

In the original exercise, we applied the distributive property to the expression \( (x + 1)(x - 3) \). First, we distribute \( x \) across \( (x - 3) \) to get \( x(x - 3) = x^2 - 3x \). Then, we distribute the 1 across \( (x - 3) \) to get \( 1(x - 3) = x - 3 \).

By applying the distributive property step-by-step, you can break down complex expressions into simpler parts that are easier to work with. It's essential to keep track of each term through this process to ensure accuracy.

Exploring Binomial Expansion

Binomial expansion, or multiplying two binomials, is closely related to the distributive property. Binomials are expressions that contain two terms, like \( (x+1) \) and \( (x-3) \). To multiply these binomials, we apply the distributive property twice, which is often remembered by the acronym FOIL: First, Outer, Inner, Last.

In our problem, we apply this method:

• First: Multiply the first terms in each binomial: \( x \cdot x = x^2 \).
• Outer: Multiply the outer terms: \( x \cdot (-3) = -3x \).
• Inner: Multiply the inner terms: \( 1 \cdot x = x \).
• Last: Multiply the last terms in each binomial: \( 1 \cdot (-3) = -3 \).

Combining these results gives: \( x^2 - 3x + x - 3 \). The concept of binomial expansion ensures each term is accounted for during multiplication, resulting in complete expansions.

Simplifying Using Like Terms

Simplification involves gathering and combining like terms, which are terms with the same variable raised to the same power. This process assists in reducing the complexity of expressions.

In the expression we expanded, \( x^2 - 3x + x - 3 \), the like terms are \( -3x \) and \( x \). Both have the variable \( x \) raised to the first power. While simplifying, these can be added together:

• The coefficient of \( -3x \) is \(-3\), and the coefficient of \( x \) is \(1\), thus combining them gives \( (-3 + 1)x = -2x \).

This results in the simplified expression: \( x^2 - 2x - 3 \). Like terms simplification transforms complex polynomials into more manageable forms. Identifying and simplifying like terms is essential for every algebraic computation and leads to streamlined expressions.