Problem 101 - Algebra A Combined Function

Question

Perform each indicated operation. \((2 x+3)(x-5)\)

Step-by-Step Solution

Answer

The simplified expression is \(2x^2 - 7x - 15\).

Step 1: Apply the Distributive Property

The exercise involves multiplying two binomials: \[(2x + 3)(x - 5)\].Apply the distributive property to multiply each term in the first binomial by each term in the second binomial. This involves four multiplications: 1. \(2x imes x\)2. \(2x imes (-5)\)3. \(3 imes x\)4. \(3 imes (-5)\).

Step 2: Perform Each Multiplication

Now, perform the calculations for each of the four terms identified:1. \(2x imes x = 2x^2\)2. \(2x imes (-5) = -10x\)3. \(3 imes x = 3x\)4. \(3 imes (-5) = -15\).

Step 3: Combine All Terms

After performing each multiplication, you have:\[2x^2, -10x, 3x, -15\].Combine these terms to form a single expression:\[2x^2 - 10x + 3x - 15\].

Step 4: Combine Like Terms

In the expression \[2x^2 - 10x + 3x - 15\], combine the like terms:- The \(x\) terms: \(-10x + 3x = -7x\).The final expression becomes:\[2x^2 - 7x - 15\].

Key Concepts

Distributive Property

The distributive property is a crucial mathematical rule when dealing with binomial multiplication. It states that for any three numbers, say a, b, and c, the expression \(a(b + c)\) can be expanded as \(ab + ac\). Essentially, each term inside the parentheses is multiplied by the term outside. This property is particularly helpful when multiplying binomials, which are expressions containing two terms, like \((2x + 3)(x - 5)\).

To use the distributive property here, consider each term in the first binomial (\(2x\) and \(3\)) and distribute these over each term in the second binomial (\(x\) and \(-5\)). This requires calculating:

• \(2x \times x\)
• \(2x \times (-5)\)
• \(3 \times x\)
• \(3 \times (-5)\)

After finding all these products, you will have a new expression containing multiple terms, which sets the stage for the next step: combining like terms.

Combining Like Terms

Combining like terms is a method used to simplify algebraic expressions. It involves merging terms in an expression that have the same variable raised to the same power. For example, in the expression \(2x^2 - 10x + 3x - 15\), the terms \(-10x\) and \(+3x\) are 'like terms' because they both involve the variable \(x\) raised to the first power.

To combine them, simply add or subtract their coefficients. Here, \(-10 + 3 = -7\). Thus, the expression simplifies to \(2x^2 - 7x - 15\).

• Identify terms with the same variable(s) and power(s).
• Add or subtract the coefficients as needed.
• Rewrite the expression with the combined terms joined.

By simplifying in this manner, expressions become easier to interpret and solve.

Polynomial Expressions

A polynomial expression is a sum of terms, each consisting of a variable raised to an integer power and multiplied by a coefficient. Polynomials can vary in complexity, containing multiple terms of differing degrees. In our exercise, the expression \(2x^2 - 7x - 15\) is a polynomial formed after multiplying two binomials and simplifying through combining like terms.

Key features of polynomial expressions include:

• Each term is in the form \(ax^n\), where \(a\) is a coefficient and \(n\) is a non-negative integer.
• The degree of a polynomial is the highest power of the variable.
• This expression has a degree of 2, indicated by the term \(2x^2\).

Understanding polynomial expressions is vital for algebraic operations as they provide the foundation for further study in calculus and other advanced math topics. They allow us to represent complex relationships and model real-world phenomena in a manageable form.