Problem 39 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ 2 x^{2}+5 x+3 $$

Step-by-Step Solution

Answer

The expression factors to \((2x + 3)(x + 1)\).

Step 1: Identify Coefficients

The quadratic trinomial is \( 2x^2 + 5x + 3 \). Identify the coefficients of the expression as follows: \( a = 2 \), \( b = 5 \), \( c = 3 \).

Step 2: Multiply and Set Up

Multiply the coefficient of \( x^2 \) (which is \( 2 \)) with the constant term (which is \( 3 \)) to get \( 2 \times 3 = 6 \). We need to find two numbers that multiply to 6 and add up to 5 (the coefficient of \( x \)).

Step 3: Find the Numbers

The numbers that multiply to 6 and sum up to 5 are 2 and 3. That is, \( 2 \times 3 = 6 \) and \( 2 + 3 = 5 \).

Step 4: Split the Middle Term

Split the middle term \( 5x \) into \( 2x + 3x \). The expression now becomes \( 2x^2 + 2x + 3x + 3 \).

Step 5: Group and Factor

Group the terms into pairs: \((2x^2 + 2x) + (3x + 3)\). Now, factor out the greatest common factor from each pair. \( 2x(x + 1) + 3(x + 1) \).

Step 6: Factor the Common Binomial

Notice \( (x + 1) \) is a common factor. Factor it out: \((2x + 3)(x + 1)\).

Step 7: Verify the Factorization

Expand \((2x + 3)(x + 1)\) to check: \(2x^2 + 2x + 3x + 3 = 2x^2 + 5x + 3\), confirming the factorization is correct.

Key Concepts

Quadratic Trinomial

A quadratic trinomial is an expression of the form \( ax^2 + bx + c \). It features three terms where the highest degree is 2, making it a quadratic equation. In this expression, the highest degree, 2, is critical as it determines that the equation is quadratic. Quadratics are important in algebra as they often appear in various mathematical problems and real-world applications. Understanding how to manipulate and factor these expressions is key for solving quadratic equations efficiently.
Quadratic trinomials can often be factored into simpler expressions called binomials. This process helps simplify solving the equation by reducing it to terms we can manage more easily.

Coefficients

Coefficients are the numerical or constant multipliers of the variables in an algebraic expression. In the quadratic trinomial \( 2x^2 + 5x + 3 \), the coefficients are identified as follows:

• \( a = 2 \) for the \( x^2 \) term
• \( b = 5 \) for the \( x \) term
• \( c = 3 \) as the constant term

Each coefficient plays a distinct role in shaping the parabola represented by the quadratic equation.
In quadratic expressions, coefficients are essential for various algebraic operations, including factoring and determining the vertex or roots of the equation. Understanding and identifying coefficients correctly is the first step in many algebraic equations tasks.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more numbers. In algebra, finding the GCF of terms in an expression allows you to factor them more easily. In our problem, we grouped the expression as \((2x^2 + 2x) + (3x + 3)\). By identifying the GCF of each group, we simplified:

• For \(2x^2 + 2x\), the GCF is \(2x\).
• For \(3x + 3\), the GCF is \(3\).

This process of extracting the GCF is crucial as it prepares the expression for further factoring into simpler binomials. It can streamline the simplification process and reduce complexity.

Binomial Factoring

Binomial factoring involves expressing a quadratic trinomial as a product of two binomials. This is demonstrated in the expression \( 2x^2 + 5x + 3 \). After grouping and finding the GCF for each set, we have the expression in the form \( 2x(x + 1) + 3(x + 1) \).
Notice how \((x + 1)\) is common in both terms. This allows us to factor it out, resulting in \((2x + 3)(x + 1)\). This step simplifies the trinomial into two manageable factors and is often a goal when solving quadratic equations for roots.
Proper binomial factoring is fundamental in algebra, making complex problems easier to handle and solutions evident. It underscores the elegance of algebraic manipulation when faced with seemingly complicated polynomials.