Question

\(y=(x+1)(x-3)\)

Short answer

The simplified form of the expression is \(y = x^2 - 2x - 3\)

Step-by-step solution

Apply the Distributive Rule (1st and Last term)

Multiply the first terms from each binomial, and the last terms. In our exercise, that will be \(x \cdot x\) and \(1 \cdot (-3)\), leading to \(x^2\) and \(-3\).

Apply the Distributive Rule (Outer and Inner term)

Multiply the outer terms and the inner terms. Here, that's \(x \cdot (-3)\) and \(1 \cdot x\) leading to \(-3x\) and \(x\).

Sum up the results

Now we need to combine the results obtained in the previous steps to get our simplified expression. This will be \(y = x^2 - 3x +x - 3\).

Key concepts

Distributive Property

The distributive property is a fundamental concept in algebra that simplifies multiplication across addition or subtraction. It's often stated as:

• \(a(b + c) = ab + ac\)

This means you distribute the multiplication of \(a\) to both \(b\) and \(c\).
In our example, we use the distributive property to multiply two binomials: \((x+1)\) and \((x-3)\).
By applying the distributive property, we multiply each term in the first binomial by each term in the second. This leads to four multiplications:

• The first term from each binomial \(x \cdot x\)
• The last terms \(1 \cdot (-3)\)
• The outer terms \(x \cdot (-3)\)
• The inner terms \(1 \cdot x\)

Mastering this rule is crucial because it is the foundation for expanding and simplifying polynomial expressions.

Binomials

A binomial is a polynomial with exactly two terms. It's an algebraic expression that has the form \(ax + b\), where \(a\) and \(b\) could be any number or constant.
In our exercise, we are dealing with two binomials: \((x+1)\) and \((x-3)\). Each binomial consists of two terms:

• The first binomial \((x+1)\) has terms: \(x\) and \(1\)
• The second binomial \((x-3)\) has terms: \(x\) and \(-3\)

When multiplying binomials, each term of the first must be multiplied by each term of the second. Understanding how to efficiently handle binomials helps in further algebraic manipulations and reduces errors while performing calculations.

Simplification

Simplification is the process of reducing an expression to its simplest form. After distributing the terms in our binomials, we obtain:

• \(x^2\) from \(x \cdot x\)
• \(-3\) from \(1 \cdot (-3)\)
• \(-3x\) from \(x \cdot (-3)\)
• \(+x\) from \(1 \cdot x\)

The next step is to combine like terms to simplify. In this case, like terms are terms with the same variable raised to the same power.
For example, \(-3x\) and \(+x\) are like terms. We combine them to make \(-2x\). Therefore, the expression becomes:

• \(x^2 - 2x - 3\)

Simplification ensures that expressions are as concise as possible, which not only makes them easier to understand but also easier to use in further calculations.