Problem 13 - Precalculus

Question

Express each polynomial in the form \(a_{n}\left(x-r_{1}\right)\left(x-r_{2}\right) \cdots\left(x-r_{n}\right)\). $$4 x^{2}+23 x-6$$

Step-by-Step Solution

Answer

The polynomial is expressed as \\( (4x - 1)(x + 6) \\).

Step 1: Identify the Expression

The given polynomial is \(4x^2 + 23x - 6\). We need to express it in factored form.

Step 2: Set Up the Factoring Format

We are looking for two numbers that multiply to the product of the leading coefficient and the constant term (\(4 \times -6 = -24\)), and add up to the middle coefficient (23).

Step 3: Find Factor Pairs

The numbers that satisfy these conditions are 24 and -1 because \(24 \times -1 = -24\) and \(24 + (-1) = 23\).

Step 4: Rewrite the Middle Term

Rewrite the polynomial as \(4x^2 + 24x - x - 6\).

Step 5: Group Terms

Group the terms into two pairs: \((4x^2 + 24x)\) and \((-x - 6)\).

Step 6: Factor by Grouping

Factor each pair: \(4x(x + 6) - 1(x + 6)\).

Step 7: Extract Common Factor

Factor out the common factor \((x+6)\), resulting in \((x + 6)(4x - 1)\).

Step 8: Verify and Write Answer

Check the factored form: \((x+6)(4x-1) = 4x^2 + 23x - 6\), which matches the original polynomial. The polynomial in factored form is expressed as \((4x - 1)(x + 6)\).

Key Concepts

Quadratic Equations

A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \). This is called the standard form of a quadratic equation, where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable or unknown. The highest degree of \( x \) in a quadratic equation is 2. A key step in solving these equations is transforming them into a factored form, which allows us to easily find the solutions, also known as roots or zeros.
Quadratic equations can be solved using several methods, such as:

• Factoring: This involves writing the quadratic in a form where it can be expressed as a product of two binomials. For example, \( ax^2 + bx + c \) can be factored into \((x - r_1)(x - r_2) \). Knowing how to identify suitable pairs of numbers to achieve this is crucial.
• Using the quadratic formula: This universal method enables solving quadratics by finding roots using the formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Completing the square: This method involves rearranging the equation to form a perfect square trinomial, thus finding the roots with greater ease.

Polynomial Expressions

Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. The key parts of a polynomial include:

• Coefficients: The numbers in front of the variables, such as 4 in \( 4x^2 \).
• Variables: Symbols like \( x \) that represent unknown quantities.
• Exponents: The power to which a variable is raised, indicating the degree of the term.

The degree of a polynomial is determined by the highest power of the variable present in it. In the given exercise, \( 4x^2 + 23x - 6 \), it's a quadratic polynomial because the highest exponent is 2.
Polynomial expressions are often solved for a specific variable value, simplified for easier computation, or factored into components for problem-solving. Factoring, like in the solution presented, is a method used to break down polynomials into simpler multiplied terms, making it easier to analyze or solve equations. It’s a vital skill in algebra, providing a foundation for more advanced mathematics.

Factored Form

The factored form of a polynomial is a representation that expresses it as the product of its factors. This is incredibly useful for simplifying expressions and solving equations. For example, the factored form of the polynomial \( 4x^2 + 23x - 6 \) is \((4x - 1)(x + 6)\).
To convert a polynomial into a factored form, you need to:

• Find factor pairs: Look for number pairs whose product equals the designated term and whose sum equals another term in the expression, just like how we identified 24 and -1 in the step-by-step solution.
• Group terms: Regroup the polynomial terms to facilitate factoring by grouping.
• Extract common factors: Identify terms appearing in multiple groups and factor them out to achieve the final form.

Putting a polynomial into its factored form helps find its roots by setting each factor to zero and solving for the variable. This ability makes factoring a fundamental skill in algebra, used to solve quadratic equations and beyond. Understanding how to convert polynomial expressions into factored form gives you a powerful tool for solving a variety of mathematical problems.