Problem 18 - Elementary and Intermediate Algebra

Question

Complete each factorization. $$ \begin{aligned} 2 y^{2}+10 y+12 &=\left(y^{2}+5 y+6\right) \\ &=2(y+\quad)(\quad+2) \end{aligned} $$

Step-by-Step Solution

Answer

2(y+3)(y+2)

Step 1: Analyze Given Equation

We start with the quadratic expression \( y^2 + 5y + 6 \). This is the expression we need to factor, which is inside the parentheses after factoring out the 2 from \( 2y^2 + 10y + 12 \).

Step 2: Find Two Numbers that Multiply to Constant Term

The goal is to factor \( y^2 + 5y + 6 \) into the form \( (y + a)(y + b) \). We need two numbers that multiply to 6 (the constant term) and add up to 5 (the coefficient of the middle term). The numbers 2 and 3 fit since \( 2 \times 3 = 6 \) and \( 2 + 3 = 5 \).

Step 3: Write the Factored Form

Using the numbers found, we write the factorization as \( y^2 + 5y + 6 = (y + 2)(y + 3) \).

Step 4: Substitute Back into Original Expression

Substitute the factorization back into the original expression: \( 2y^2 + 10y + 12 = 2(y + 2)(y + 3) \).

Step 5: Complete the Remaining Factorization

Fill in the blanks to complete the factorization: \( 2(y + 3)(y + 2) \). We can interchange \( y + 2 \) and \( y + 3 \) since multiplication is commutative.

Key Concepts

Quadratic Expressions

Quadratic expressions are an essential part of algebra and commonly appear in various kinds of mathematical problems. They are polynomials of the second degree, usually written in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. The quadratic expression features a squared term, a linear term, and a constant term.

In the context of the exercise, the given quadratic expression is \( y^2 + 5y + 6 \). It includes:

• A squared term: \( y^2 \)
• A linear term: \( 5y \)
• A constant term: 6

Understanding the components of a quadratic expression is crucial as it sets the stage for factorization. Factorization helps in simplifying and solving quadratic expressions, crucial for many applications in mathematics.

Factorization Steps

Factorization is a technique used to express a polynomial as a product of its factors. The process of factoring a quadratic expression involves specific steps:

• Identify the Expression: Recognize the quadratic expression you need to factor. For instance, starting with \( y^2 + 5y + 6 \) from the exercise.
• Find Two Numbers: These numbers should multiply to give you the constant term (6 in this case) and add to give you the coefficient of the linear term (5). Here, the numbers 2 and 3 do the job as they satisfy both conditions: \( 2 \times 3 = 6 \) and \( 2 + 3 = 5 \).
• Write Factored Form: Using these numbers, rewrite the expression as a product of two binomials: \( (y + 2)(y + 3) \).

Once the polynomial is factored, you can use it to simplify expressions or solve equations.

Polynomial Factorization

Polynomial factorization simplifies expressions and unravels solutions to polynomial equations. In the exercise, we factored a quadratic polynomial, which involves expressing it as the product of linear polynomials.

The polynomial begins as \( 2y^2 + 10y + 12 \). Initially, common factors like 2 are factored out to simplify the process, reducing it to finding the factors of \( y^2 + 5y + 6 \).

When factorizing:

• First, check for a greatest common factor (GCF) across all polynomial terms. Here, it's 2, simplifying our job.
• Next, focus on the resultant quadratic and apply the factorization into linear terms methodically.
• Finish by reassembling the expression using the factors discovered, leading to the complete factorization: \( 2(y + 2)(y + 3) \).

The process of polynomial factorization is invaluable in algebra, like solving quadratic equations, simplifying expressions, and understanding relationships between roots and coefficients.