Problem 63 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(x^{2}+4 x+3\)

Step-by-Step Solution

Answer

.

Step 1 - Identify coefficients

Identify the coefficients for the given trinomial. For the trinomial in the form of , we have: .

Step 2 - Find factors of the constant term

Find pairs of factors that multiply to the constant term . In this case, we are looking for pairs of numbers that multiply to 3, which are: (1, 3) and (-1, -3).

Step 3 - Determine the correct pair of factors

From the pairs found, determine the pair of factors that add up to the middle coefficient . For , the correct pair is (1, 3) since .

Step 4 - Write the factored form

Now express the trinomial as a product of two binomials. Using the pair (1, 3), we write the factored form as: .

Key Concepts

elementary algebra

Elementary algebra provides foundational tools for understanding and solving equations. A core element is the manipulation of expressions involving variables. When we work with trinomials, we particularly focus on expressions that follow the form \(ax^2 + bx + c\). Understanding how to break these expressions down and simplify them through techniques like factoring is essential.

Elementary algebra involves:

• Identifying terms and coefficients
• Performing algebraic operations like addition, subtraction, multiplication, and division with variables
• Solving for unknowns

In our exercise, identifying the coefficients of the trinomial \(x^2 + 4x + 3\) and using them effectively is key. The coefficients are the numerical values next to each term (\(a\), \(b\), and \(c\)).

factorization

Factorization is breaking down an expression into 'factors' that, when multiplied, give the original expression. For trinomials \(ax^2 + bx + c \), this means expressing it as a product of two binomials. The steps include:

• Identifying the constant and middle terms
• Finding pairs of numbers that multiply to the constant term (\(c\))
• Selecting the pair that adds up to the middle term (\(b\))

In our trinomial \(x^2 + 4x + 3\), we identify pairs of factors for 3—like (1, 3) and (-1, -3). We select (1, 3) since they add up to 4. This helps us write the expression as a product of binomials: \(x^2 + 4x + 3 = (x + 1)(x + 3)\).

polynomials

Polynomials are algebraic expressions with multiple terms, typically written in descending power order. These can include constants, variables, and their exponents—like \(x^3 + 2x^2 - 4x + 7\).

A polynomial of degree 2, often called a quadratic polynomial, is written in the form \(ax^2 + bx + c\) (like our example). Understanding the components:

• \(a, b, \text{and} c\) are coefficients
• Two variable terms (\(x^2\) and \(x\)) and a constant term

Factoring helps simplify polynomial expressions by breaking them into products of simpler polynomials or binomials, making them easier to solve or manipulate in equations.