Question

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

Short answer

Question: Simplify the given expression: \((x+1)(x-3)(x-1)\) Answer: \(x^3 - 3x^2 - x + 3\)

Step-by-step solution

Expand the first two terms

First, expand the \((x+1)(x-3)\) part of the expression by applying the distributive property (FOIL method). Multiply the first terms, the outer terms, the inner terms, and the last terms, then add them up: \((x+1)(x-3) = (x \cdot x) + (-3 \cdot x) + (1 \cdot x) + (1 \cdot -3)\). This simplifies to \(x^2 - 2x - 3\).

Expand the result with the third term

Next, expand the result from Step 1 with the remaining term, \((x-1)\). Multiply each term from the result of Step 1 with each term from the \((x-1)\): \((x^2-2x-3)(x-1)\). Apply the distributive property (FOIL method) again: \((x^2-2x-3)(x-1) = (x^2 \cdot x) + (-1 \cdot x^2) + (-2x \cdot x) + (2x \cdot 1) + (-3 \cdot x) + (3 \cdot 1)\).

Combine like terms

Combine the like terms from the expression in Step 2: \(x^3 - x^2 - 2x^2 + 2x - 3x + 3 = x^3 - 3x^2 - x + 3\). This is the simplified form of the given expression after removing the brackets. So the final answer is: \(x^3 - 3x^2 - x + 3\).

Key concepts

Distributive Property

The distributive property is a foundational concept in algebra, especially essential for polynomial expansion. It involves distributing each term in one set of parentheses across each term in another. This technique helps in simplifying expressions and is frequently used to tackle problems where terms are multiplied.

• The Formula: The distributive property formula is written as:
  \(a(b + c) = ab + ac\).
• Application: For expressions like \((x+1)(x-3)\), distribute each term in \((x+1)\) to each term in \((x-3)\) to expand it.

By using the distributive property, every term from the first binomial gets multiplied by each term in the second. This results in a new polynomial which is then further simplified by combining the like terms.

FOIL Method

The FOIL method is a mnemonic that stands for First, Outer, Inner, Last. It is specifically used for multiplying two binomials, helping you remember the steps for distribution:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms in the expression.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

In the given problem, we use the FOIL method to expand \((x+1)(x-3)\) by following these steps:

• First: \(x \cdot x = x^2\)
• Outer: \(x \cdot (-3) = -3x\)
• Inner: \(1 \cdot x = x\)
• Last: \(1 \cdot -3 = -3\)

After applying the FOIL method, the binomial multiplication results in the expression \(x^2 - 2x - 3\). This method is a step to ensure no part of the binomial gets left out during multiplication.

Combining Like Terms

Combining like terms is the process of simplifying expressions by summing the coefficients of terms that have the same variables and exponents. It helps reduce an expression to its simplest form. During polynomial expansion, many terms can be combined to achieve a more compact expression.

• Same Variables and Exponents: Like terms are terms that have identical variables raised to the same power (e.g., \(2x^2\) and \(-3x^2\)).
• Simplification: Combining them involves adding or subtracting their coefficients.

In our exercise:

• Combine \(-x^2\) with \(-2x^2\) to get \(-3x^2\).
• Combine \(2x\) and \(-3x\) to get \(-x\).

This practice ensures that all possible simplifications are done, resulting in a clean and concise expression \(x^3 - 3x^2 - x + 3\). It's a critical skill for polynomial equations, streamlining them into a form that is often much easier to interpret and solve.