Problem 21 - Elementary Algebra

Question

For the following problems, factor the trinomials when possible. $$ y^{2}+10 y+16 $$

Step-by-Step Solution

Answer

Answer: The factored form of the trinomial \(y^2 + 10y + 16\) is \((y + 8)(y + 2)\).

Step 1: Identify the trinomial form

The given equation is a quadratic trinomial in standard form: \(y^2 + 10y + 16\).

Step 2: Find two numbers that multiply to the constant term and sum to the linear term coefficient

We need to find two numbers that multiply to 16 and add up to 10. These two numbers are 8 and 2, since \(8 * 2 = 16\) and \(8 + 2 = 10\).

Step 3: Factor the trinomial

Now that we have found the two numbers (8 and 2), we can write the trinomial as a product of two binomials. Replace the linear term \(10y\) with the sum of \(8y\) and \(2y\) and then factor by grouping. $$ y^2 + 8y + 2y + 16 $$ Group the terms: $$ (y^2 + 8y) + (2y + 16) $$ Factor out the common factors in each group: $$ y(y + 8) + 2(y + 8) $$ Factor out the common binomial \((y + 8)\): $$ (y + 8)(y + 2) $$

Step 4: Write the final factored form

The factored form of the given trinomial \(y^2 + 10y + 16\) is: $$ (y + 8)(y + 2) $$

Key Concepts

Quadratic Trinomial

When we talk about a quadratic trinomial, we're referring to a polynomial with three terms. The standard form of a quadratic trinomial is typically represented as ax^2 + bx + c, where a, b, and c are constants, and x represents the variable. The 'quadratic' part means the highest power of the variable is 2.

Quadratic trinomials are frequently encountered in algebra and other areas of mathematics because they describe parabolic shapes when graphed. Factoring a quadratic trinomial essentially means breaking it down into simpler, multiplicative components. For instance, the trinomial y^2 + 10y + 16 will be factored into two binomials. This process is vital as it aids in solving quadratic equations, simplifying expressions, and analyzing function properties such as roots and intercepts.

Factor by Grouping

The technique known as factor by grouping is a method used to factor certain kinds of polynomials, including some quadratic trinomials. It involves rearranging and grouping terms in the polynomial so that we can factor out common factors.

To apply this method, we first look for a way to split the middle term into two terms whose coefficients can help us group all the terms into pairs. Each pair will have a common factor that can be factored out. After this step, if successful, a common binomial factor will emerge, which can then be factored out to reveal the product of two binomials. For instance, in the trinomial y^2 + 10y + 16, by replacing the term 10y with 8y + 2y and grouping, we can factor by grouping to obtain the outcome (y + 8)(y + 2).

Binomial Factors

In the context of factoring quadratic trinomials, binomial factors are essentially the building blocks that make up the expression when it's fully factored. A binomial is a polynomial with just two terms. When we factor a trinomial, we are looking to express it as the product of two binomial factors.

For example, the quadratic trinomial y^2 + 10y + 16, when factored, gives us the binomial factors (y + 8) and (y + 2). Each of these binomials is simpler than the original trinomial, and when multiplied together, they give back the original trinomial. Factoring into binomials is especially useful for solving quadratic equations, where setting each binomial equal to zero can provide the solutions to the equation represented by the original trinomial.