Problem 65 - Elementary Algebra

Question

For the following problems, factor, if possible, the trinomials. $$ y^{2}+20 y+100 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the terms

In the given trinomial $$ y^{2} + 20y + 100, $$ we have the following terms: - A quadratic term: \(y^2\) - A linear term: \(20y\) - A constant term: \(100\)

Step 2: Find factors of the constant term

Now, let's find the factors of the constant term, 100, that add up to the coefficient of the linear term (20): 1. (1, 100) = 1 + 100 = 101 2. (2, 50) = 2 + 50 = 52 3. (4, 25) = 4 + 25 = 29 4. (5, 20) = 5 + 20 = 25 5. (10, 10) = 10 + 10 = 20 We found the required factors, which are (10, 10).

Step 3: Write the trinomial as a product of two binomials

Now that we have the factors, we can rewrite the trinomial as a product of two binomials. Using the factors we found in step 2, we can write the binomials as \((y + 10)(y + 10)\).

Step 4: Simplify the product of the binomials (if possible)

Since the two binomials are the same, we can simplify the expression as follows: $$ (y + 10)(y + 10) = (y + 10)^2. $$

Step 5: Conclusion

The factored form of the given trinomial \(y^2 + 20y + 100\) is \((y + 10)^2\).

Key Concepts

Quadratic Equations

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). It’s characterized by the highest exponent, which is 2, indicating it’s a second-degree polynomial. Different methods can be used to solve quadratic equations, including factoring, completing the square, and using the quadratic formula.

In the context of the exercise, we look at a quadratic expression \( y^2 + 20y + 100 \). Although it’s not set equal to zero, recognizing it as a quadratic form helps in understanding how to factor it. Identifying it in standard form means understanding the roles of each term: \( y^2 \) is the quadratic term, \( 20y \) is the linear term, and \( 100 \) is the constant term.

Quadratic equations often appear in a variety of real-world scenarios, like projectile motion in physics or optimizing areas in geometry, which highlights their importance in different fields of study.

Factoring Techniques

Factoring is a method used to express an algebraic expression as the product of its factors. This skill is vital when working with polynomial equations, especially quadratics. Factoring simplifies equations and makes them more manageable to solve.

For the given trinomial \( y^2 + 20y + 100 \), the focus is on finding two numbers that both add up to the middle term and multiply to the constant term 100. Here’s how this works:

• List pairs of numbers that multiply to 100.
• Test the pairs to find which ones add to 20, the coefficient of the middle term.
• Once found, rewrite the expression into binomials: \((y + 10)(y + 10)\).

This is known as the "factoring by grouping" and "trial and error" method, providing a practical approach for students learning basic factoring techniques.

Simplifying these binomials can further highlight that \( (y + 10)(y + 10) = (y + 10)^2 \), an expression more visually pleasant and neatly compressed. Factoring techniques like these are essential tools in not just algebra, but in calculus, data science, and other mathematical applications.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations, forming the building blocks of algebra. They can represent actual quantities and are foundational in formulating equations and inequalities.

In the trinomial \( y^2 + 20y + 100 \), understanding what makes up an algebraic expression is crucial: the expression includes a variable \( y \), raised to different powers, combined with constants through addition. This formulation assists students in recognizing and manipulating similar expressions.

Breaking down algebraic expressions into simpler components, such as through factoring, helps in solving equations, modeling scenarios, and even in computer algorithms. With regular practice, comprehending and transforming algebraic expressions becomes intuitive, aiding not only in standardized tests but in everyday problem-solving tasks.