Problem 77 - Algebra A Combined Function

Question

Write a polynomial that factors as \((x-3)(x+8)\).

Step-by-Step Solution

Answer

The polynomial is \( x^2 + 5x - 24 \).

Step 1: Identify the form of the polynomial

To find the polynomial from its factors, understand that it is written in the form \( (x-a)(x-b) \). Here, \(a = 3\) and \(b = -8\).

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Step 6: Conclude with the answer

The polynomial is \( x^2 + 5x - 24 \).

Key Concepts

Polynomial Equations

Polynomial equations are fascinating tools in mathematics that involve expressions with variables raised to whole number powers. The solutions of these equations hold great significance in various fields of study and real-world problems. A polynomial equation is generally written in the form:

• A polynomial equation might look like this: \( ax^n + bx^{n-1} + \cdots + k = 0 \)
• 'n' must be a non-negative integer, and 'a', 'b', etc., are coefficients.
• The highest power of the variable, known as the degree of the polynomial, determines the number of solutions.

For example, a quadratic polynomial equation like \((x-3)(x+8) = 0\) is solved by setting each factor equal to zero and solving for 'x'. This results in the solutions, or 'roots', of the equation. Each root represents a value that makes the polynomial equation equal to zero.

Algebraic Expressions

Algebraic expressions form the backbone of polynomial equations. These expressions consist of variables, constants, and arithmetic operations like addition, subtraction, multiplication, and division.

• An example of an algebraic expression is \(x - 3\), where 'x' is the variable, and '3' is a constant.
• These expressions can be combined in various ways to form polynomials of different degrees.

When you create a polynomial from its factored form, you are essentially expanding these algebraic expressions. For instance, the expression \((x-3)(x+8)\) is an algebraic way to represent a polynomial. By multiplying these binomials, we can transform it into a standard polynomial form.

Multiplying Binomials

Multiplying binomials is a fundamental skill in algebra that allows you to expand expressions into polynomial form. Each binomial is a simple polynomial expression with two terms, like \((x-3)\) or \((x+8)\). The process of multiplying binomials involves the use of the distributive property.

• Consider the multiplication \((x-3)(x+8)\). This requires distributing each term in the first binomial across each term in the second binomial.
• First, multiply \(x\) by each term in \((x+8)\), resulting in \(x^2 + 8x\).
• Then, multiply \(-3\) by each term in \((x+8)\), which results in \(-3x - 24\).
• Combine all terms to get the expanded polynomial: \(x^2 + 5x - 24\).

This method, often referred to as the FOIL method when working with two binomials, stands for First, Outer, Inner, Last, representing the order of the terms multiplied in binomials. Understanding and mastering this technique is crucial for simplifying and solving more complex polynomial problems.