91Ó°ÊÓ

The distributive property is a fundamental concept in algebra that is especially handy when multiplying binomials. In essence, this property allows us to multiply each term inside a parenthesis by a term outside the parenthesis. This is expressed as: \[ a(b + c) = ab + ac \] For the expression \((x - 5)(x + 7)\), the distributive property is applied twice. First, we distribute \(x\) across \((x + 7)\), resulting in \(x^2 + 7x\). Then, distribute \(-5\) across \((x + 7)\), resulting in \(-5x - 35\). We then add together all the terms to get \(x^2 + 2x - 35\).
Remember:

• Distribute each term in the first binomial across the second binomial.
• Combine like terms after distributing.

This property simplifies the task of multiplying each term separately and makes it easier to handle complex algebraic expressions.

Binomial multiplication refers to multiplying two binomial expressions. A binomial is simply an algebraic expression containing two terms, such as \((x - 5)\) or \( (x + 7)\). The process uses well-known patterns like FOIL or the distributive property to simplify multiplication. In our example, \((x - 5)(x + 7)\), you apply binomial multiplication as follows:

• Identify parts: \(a=x, b=-5, c=7\).
• Use the formula for multiplying binomials: \[ (a+b)(a+c) = a^2 + (b+c)a + bc \].
• Calculate each term step by step.

Unlike simple multiplication, binomial multiplication involves multiple terms which must be carefully combined. Practice helps in learning patterns or shortcuts to easily manage these multiplications. Applying these methods helps solve algebraic problems efficiently.

Algebraic expressions are a combination of variables, numbers, and operators. They are fundamental in forming algebraic equations and inequalities. Understanding them is key to mastering algebra. In the exercise \((x - 5)(x + 7)\), which involves binomials, we deal with variables and constants:

• Variables: Symbols like \(x\) that represent numbers. They can change values.
• Constants: Fixed numbers \(-5\) and \(7\) that do not change.
• Operators: Symbols such as \( + \), \( - \), and \( \times \) that indicate which mathematical operations to perform.

Algebraic expressions can be manipulated using operations like addition, subtraction, multiplication, and division. Learning how to rewrite and simplify these expressions is crucial. The eventual goal in many cases is to find a simplified or standard form, such as rewriting \((x-5)(x+7)\) as \(x^2 + 2x - 35\). Practice can strengthen your grasp of manipulating algebraic expressions and solving equations.