Problem 31 - Elementary and Intermediate Algebra

Question

Add the polynomials. $$ \left(\frac{2}{3} y^{3}+\frac{3}{4} y^{2}+\frac{1}{2}\right)+\left(\frac{1}{3} y^{3}+\frac{1}{5} y^{2}-\frac{1}{6}\right) $$

Step-by-Step Solution

Answer

The result is \(y^3 + \frac{19}{20}y^2 + \frac{1}{3}\).

Step 1: Write Down the Expression

First, write down the given expression clearly: \[\left(\frac{2}{3}y^{3} + \frac{3}{4}y^{2} + \frac{1}{2}\right) + \left(\frac{1}{3}y^{3} + \frac{1}{5}y^{2} - \frac{1}{6}\right)\]

Step 2: Group Like Terms

Identify and group the like terms together, which are terms with the same powers of \(y\):\[\left(\frac{2}{3}y^{3} + \frac{1}{3}y^{3}\right) + \left(\frac{3}{4}y^{2} + \frac{1}{5}y^{2}\right) + \left(\frac{1}{2} - \frac{1}{6}\right)\]

Step 3: Simplify Each Group

Simplify each group of like terms by adding the coefficients:- For \(y^3\) terms: \[\frac{2}{3} + \frac{1}{3} = \frac{3}{3} = 1\]- For \(y^2\) terms:\[\frac{3}{4} + \frac{1}{5} = \frac{15}{20} + \frac{4}{20} = \frac{19}{20}\]- For constant terms:\[\frac{1}{2} - \frac{1}{6} = \frac{3}{6} - \frac{1}{6} = \frac{2}{6} = \frac{1}{3}\]

Step 4: Combine Simplified Terms

Combine all the simplified terms to form the final polynomial:\[y^3 + \frac{19}{20}y^2 + \frac{1}{3}\]

Key Concepts

Like Terms

Understanding like terms is the foundation for manipulating algebraic expressions. In algebra, like terms are terms that have identical variable parts and powers. For example, in the polynomial expression provided, terms with the variable part of \(y^3\) and \(y^2\) are like terms because they share the same variable and exponent.

Identifying like terms is crucial because it allows you to group them together for simplification. Grouping makes it easier to perform operations such as addition or subtraction on the coefficients. As seen in the exercise, terms like \(\frac{2}{3}y^3\) and \(\frac{1}{3}y^3\) are combined because they are both raised to the third power of \(y\). Similarly, \(\frac{3}{4}y^2\) and \(\frac{1}{5}y^2\) are grouped as they both involve \(y^2\). Also, don't forget the constant terms, which are like terms as well, independent of variables.

• Identical variables and powers define like terms.
• Grouping makes computations streamlined.
• Simplifying like terms involves operating on the coefficients.

Once you’ve grouped like terms, you’re set up to simplify the expression further, making your math work more manageable.

Simplify Coefficients

After identifying like terms, the next major step is simplifying the coefficients. Coefficients are the numerical parts of terms. In this exercise, they include fractions like \(\frac{2}{3}\) and \(\frac{1}{5}\).

Simplifying coefficients involves performing arithmetic operations on these numbers. It's akin to regular number addition or subtraction but can involve fractions, making it slightly tricky. In our solution, we see these operations clearly:

For \(y^3\) terms:

• Add \(\frac{2}{3} + \frac{1}{3}\) to get \(\frac{3}{3} = 1\).

For \(y^2\) terms:

• Add \(\frac{3}{4} + \frac{1}{5}\) by finding a common denominator:
  \(\frac{15}{20} + \frac{4}{20} = \frac{19}{20}\).

For constant terms:

• Subtract \(\frac{1}{6}\) from \(\frac{1}{2}\):
  \(\frac{3}{6} - \frac{1}{6} = \frac{2}{6} = \frac{1}{3}\).

Working with fractions entails finding a common denominator for addition or subtraction, which is fundamental in simplifying expressions with fractional coefficients. This process transforms a complicated-looking equation into something much more manageable and clear.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations such as addition, subtraction, multiplication, or division. In the exercise given, the expression \( \left(\frac{2}{3}y^3 + \frac{3}{4}y^2 + \frac{1}{2}\right) + \left(\frac{1}{3}y^3 + \frac{1}{5}y^2 - \frac{1}{6}\right) \) is an excellent example of working with polynomials, a type of algebraic expression.

Polynomials can range from simple monomials like \(3x\) to complex expressions involving multiple terms, as seen here. Most importantly, to correctly simplify or solve polynomials, you need to:

• Understand and identify the like terms involved.
• Carefully work through simplifying their coefficients, particularly when dealing with fractions.
• Combine the simplified terms to form a more concise expression, as shown by our final result, \(y^3 + \frac{19}{20}y^2 + \frac{1}{3}\).

These expressions allow for versatile mathematical modeling of real-world scenarios, making the process of learning how to manipulate them a valuable skill in mathematics.