Problem 121 - Elementary Algebra

Question

Factor each expression. \(y^{2}+10 y+15\)

Step-by-Step Solution

Answer

Expression is not easily factorable with integers.

Step 1: Identify Coefficients

Identify the coefficients in the quadratic expression. For the given expression, the coefficients are:- A (coefficient of \(y^2\)) = 1- B (coefficient of y) = 10- C (constant term) = 15

Step 2: Find Factor Pairs of C

Find pairs of factors of the constant term C (15) that add up to the coefficient B (10). The pairs of factors of 15 are (1, 15) and (3, 5). Check if these pairs add up to 10:1 + 15 = 16 (not 10)3 + 5 = 8 (not 10)

Step 3: Re-examine Factorization

Since none of the factor pairs add up to 10, recall that the given quadratic expression is not easily factorable using integers. This expression can be considered as already simplified.

Key Concepts

Coefficients

In a quadratic expression, coefficients play a significant role. They are the numbers in front of the variables. In the expression given, the coefficients are the numbers attached to the terms with the variable y.
Consider the expression \( y^{2} + 10y + 15 \). We can identify the coefficients as follows:

• A (coefficient of \( y^2 \)) = 1
• B (coefficient of y) = 10
• C (constant term) = 15

These coefficients are crucial since they guide us in the process of factoring the quadratic expression. Finding the correct numbers that relate as coefficients is the first step towards factorization.

Constant Term

The constant term is another important element in a quadratic expression. It is the term without any variables, and it's the value that remains constant across the expression.
In the expression \( y^{2} + 10y + 15 \), the constant term is 15.
The constant term is essential when finding factors of the quadratic expression. We look for pairs of numbers that multiply to give the constant term. In this case, we must find pairs that multiply to 15. This step is vital as it can help us break down the expression further if factorable.

Factor Pairs

Factor pairs are sets of two numbers that, when multiplied, result in a given number. For a quadratic expression, we focus on the factor pairs of the constant term.
Looking at \( y^{2} + 10y + 15 \), we need to find the factor pairs of 15:

• (1, 15)
• (3, 5)

Next, we check if these pairs can add up to the coefficient of the middle term (10). Unfortunately, neither 1 + 15 = 16 nor 3 + 5 = 8 matches the coefficient 10.
Since none of the pairs add up to the middle coefficient, this quadratic expression can't be simplified using integer factor pairs.

Simplification

Simplification is the process of rewriting an expression in its most basic or easily understandable form. For quadratic expressions, this often involves factoring them if possible.
In our case with \( y^{2} + 10y + 15 \), after checking possible pairs of factors and finding none that sum up to the middle coefficient, we conclude it is not easily factorable.
Therefore, we consider this expression simplified at its current form. It's beneficial to learn that not all quadratics can be simplified by factoring, highlighting the expression's already simplified state.