Problem 11 - Algebra 2 and Trigonometry

Question

In \(3-12,\) write the sum or difference of the given polynomials in simplest form. $$ \left(y^{2}-y-7\right)+\left(3-2 y+3 y^{2}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(4y^2 - 3y - 4\).

Step 1: Identify Like Terms

First, let's identify like terms in the polynomials. We have two polynomials given:1. \(y^2 - y - 7\)2. \(3 - 2y + 3y^2\)Like terms here are the terms with the same power of \(y\).

Step 2: Group the Like Terms Together

Now, let's group like terms from both polynomials together:- The \(y^2\) terms: \(y^2\) and \(3y^2\)- The \(y\) terms: \(-y\) and \(-2y\)- The constant terms: \(-7\) and \(3\)

Step 3: Add the Like Terms

Now add the coefficients of like terms:- For \(y^2\) terms: \(1y^2 + 3y^2 = 4y^2\)- For \(y\) terms: \(-1y - 2y = -3y\)- For constant terms: \(-7 + 3 = -4\)

Step 4: Write the Simplified Expression

Combine all the simplified terms to form a single polynomial:\[4y^2 - 3y - 4\]

Key Concepts

Understanding Like Terms

Like terms are essential in simplifying polynomial expressions. But what makes terms "like"? Simply put, like terms are terms within a polynomial that have exactly the same variables raised to the same powers. Only the coefficients, which are the numbers in front of these variables, can be different.

In our exercise, the terms like \(y^2\) from both polynomials are like terms because they both have the variable \(y\) raised to the second power. Similarly, the terms that contain \(y\) and no exponent, \( -y \) and \(-2y\), are also like terms.

This concept is crucial because it allows us to combine these terms by adding or subtracting their coefficients. Identifying like terms correctly is the first significant step in polynomial addition or subtraction.

Simplifying Polynomials

Simplifying polynomials involves combining like terms to reduce the complexity of a polynomial expression. This process makes the expression more straightforward to understand and work with. By doing so, you transform a potentially long expression into its shortest possible form without changing its value.

To simplify a polynomial, you perform operations on like terms. This process involves:

• Grouping terms that have the same variable raised to the same power.
• Adding or subtracting the coefficients of these like terms.

In our solved exercise, each type of like term, such as \(y^2\), \(y\), and the constant, were grouped and simplified separately. For example, adding the \(y^2\) terms \(1y^2\) and \(3y^2\) provided \(4y^2\). By simplifying polynomials, we make it easier to understand their structure and solve potential equations involving them.

Exploring Polynomial Expressions

Polynomial expressions are algebraic expressions made up of terms that are combined through addition, subtraction, multiplication, or division. These terms consist of variables raised to whole-number exponents and their attached coefficients.

For example, our given polynomials \(y^2 - y - 7\) and \(3 - 2y + 3y^2\) are expressions with three terms each, where the terms include squares, linear terms, and constants.

Polynomial expressions are foundational in algebra and can represent various relationships in mathematics and real-world scenarios. They're used in equations to model everything from simple tasks, like calculating areas, to more complex phenomena, like predicting financial trends. Mastery of polynomial expressions paves the way for success in algebra and beyond, making them an indispensable part of mathematical education.