Problem 72 - Elementary Algebra

Question

Write out your own list of steps for factoring a trinomial of the form \(x_{2}+b x+c\) and share your steps on the discussion board.

Step-by-Step Solution

Answer

Factor by finding two numbers that multiply to \(c\) and add to \(b\), then write as \((x+m)(x+n)\).

Step 1: Identify and Write Down the Coefficients

For the trinomial in the form \(x^2 + bx + c\), identify the coefficients. Here, \(b\) is the coefficient of \(x\) and \(c\) is the constant term. Make sure these values are clearly noted.

Step 2: Find Two Numbers That Multiply to c

Look for two numbers that multiply to the constant term \(c\). These two numbers also need to add up to the coefficient \(b\). This is a key step for the correct factorization.

Step 3: Determine the Correct Pair

Once you have the possible pairs that multiply to \(c\), determine which pair of numbers adds up to \(b\). Verify your choice by ensuring both conditions (multiplying to \(c\) and adding to \(b\)) are met.

Step 4: Write the Factored Form

Using the selected pair of numbers, write the factored form of the trinomial as \((x + m)(x + n)\), where \(m\) and \(n\) are the numbers found in Step 3.

Step 5: Verify Your Solution

Expand the factored form \((x + m)(x + n)\) to ensure it matches the original trinomial. This step guarantees that the selected factors are correct and that the trinomial is properly factored.

Key Concepts

Quadratic Equations

Quadratic equations are algebraic expressions of the form \(ax^2 + bx + c = 0\). Here, \(a, b,\) and \(c\) represent constants where \(a eq 0\), ensuring the equation includes a squared term. The term "quadratic" comes from the Latin word "quadratus," meaning "square." This is because the highest exponent in the equation is two. These equations are fundamental in algebra, as they can model various real-world phenomena, like projectile motion.

Understanding quadratic equations involves recognizing their components:

• The \(x^2\) term, which gives the equation its name
• The \(x\) term, representing a linear aspect
• The constant term, which stands alone without \(x\)

Quadratic equations can be solved using different methods: factoring, completing the square, or applying the quadratic formula \((-b \pm \sqrt{b^2 - 4ac})/(2a)\). Factoring is often the simplest, making it important to master how to factor trinomials.

Polynomial Factoring

Polynomial factoring is a key skill in algebra. It involves expressing a polynomial as a product of simpler polynomials. This technique simplifies problems, particularly when solving equations or simplifying algebraic expressions. For trinomials of the form \(x^2 + bx + c\), the goal of factoring is to break it down into the product of two binomials, \((x + m)(x + n)\).

Here's a simplified approach to polynomial factoring:

• Identify the coefficients: Start by noting the values of \(b\) and \(c\).
• Determine factors: Find two numbers that multiply to \(c\) and add up to \(b\).
• Verify the pair: Ensure your selected numbers satisfy both conditions.
• Write the factors: Use the pair to express the trinomial as \((x + m)(x + n)\).

After factoring, always expand the binomials to verify they match the original trinomial. This step is crucial for accuracy.

Algebraic Expressions

Algebraic expressions include variables, constants, and operations (like addition or multiplication). These expressions form the foundation of many mathematical concepts, including equations and inequalities. Algebraic expressions can vary from simple, like \(3x + 2\), to complex with multiple variables and terms.

Key points about algebraic expressions:

• Variables: Symbols like \(x\) or \(y\) represent unknown values.
• Constants: These are fixed values like 1, 3, or 10 in an expression.
• Operations: Include addition (+), subtraction (-), multiplication (*), and division (/).

When working with algebraic expressions, it's critical to manipulate them correctly through techniques like factoring, substitution, or simplification. These skills help solve equations, graph functions, and interpret mathematical relationships. Mastering algebraic expressions paves the way for advanced topics in mathematics.