Problem 71 - Elementary Algebra

Question

Create your own trinomial of the form \(x_{2}+b x+c\) that factors. Share it along with the solution on the discussion board.

Step-by-Step Solution

Answer

The trinomial is \(x^2 + 5x + 6\), which factors to \((x + 2)(x + 3)\).

Step 1: Choose Coefficients

Select values for \(b\) and \(c\) in the trinomial \(x^2 + bx + c\). Let's choose \(b = 5\) and \(c = 6\). This gives us the trinomial \(x^2 + 5x + 6\).

Step 2: Verify Factorability

We need to check if the trinomial \(x^2 + 5x + 6\) can be factored. To factor, the numbers that multiply to \(c = 6\) should add up to \(b = 5\). The pairs of factors of 6 are \((1, 6)\) and \((2, 3)\). The pair \((2, 3)\) adds up to 5, so it's factorable.

Step 3: Factor the Trinomial

Using the pair \((2, 3)\), factor \(x^2 + 5x + 6\) into \((x + 2)(x + 3)\).

Step 4: Verify the Solution

Expand \((x + 2)(x + 3)\) to ensure it equals the original trinomial \(x^2 + 5x + 6\):\[ (x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6. \]The factors are correct.

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical phrases that include variables, coefficients, and operations like addition, subtraction, multiplication, and non-negative exponentiation. This means they are composed of terms that are combined to form expressions. The degree of a polynomial expression indicates the highest power of the variable present in the expression.

For example, the expression \(x^2 + 5x + 6\) is a polynomial expression where the degree is 2 because the highest exponent of \(x\) is 2.
Polynomial expressions can be classified based on their degree:

• Constant polynomial: degree 0, e.g., \(7\)
• Linear polynomial: degree 1, e.g., \(3x + 2\)
• Quadratic polynomial: degree 2, e.g., \(x^2 + 5x + 6\)

Recognizing such expressions is foundational in algebra, allowing for operations like addition and factoring.

Factoring Techniques

Factoring techniques are strategies used to decompose a polynomial into a product of simpler expressions, known as factors. When factoring polynomials, the goal is to simplify the expression so that it is easier to work with in equations and other algebraic operations.

Trinomial factoring is one common technique, especially for quadratic expressions of the form \(x^2 + bx + c\). To factor such a trinomial:

• Identify two numbers that multiply to \(c\) and add up to \(b\).
• In the example \(x^2 + 5x + 6\), the numbers 2 and 3 work because \(2 \cdot 3 = 6\) and \(2 + 3 = 5\).
• Use these numbers to express the trinomial as \((x + m)(x + n)\), where \(m\) and \(n\) are the numbers found.

This technique simplifies solving quadratic equations and understanding their properties, such as roots and vertex.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operators that represent values in algebra. These expressions form the backbone of algebraic operations, allowing for abstraction and manipulation of numbers in equations and inequalities.

When dealing with algebraic expressions like \(x^2 + 5x + 6\), letter variables stand in for numbers, adding a layer of generality and flexibility. This aids in expressing a range of possibilities rather than a fixed number.

In algebraic manipulation, expressions can be simplified, factored, or expanded to reveal useful properties or solve equations.

• Factored form, such as \((x + 2)(x + 3)\), highlights solutions or roots, where the expression equals zero.
• To expand, you distribute the terms across one another, a skill useful in verification, as shown when confirming the factors of a trinomial.

By understanding and practicing with algebraic expressions, solving complex mathematical problems becomes more achievable.