Problem 42 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ \left(5 y^{2}-3 y-1\right)-\left(2 y^{2}+y+1\right) $$

Step-by-Step Solution

Answer

\(3y^{2} - 4y - 2\)

Step 1: Identify the Expression

We are given the problem: \( \left(5y^{2} - 3y - 1\right) - \left(2y^{2} + y + 1\right) \). Notice that we are subtracting two polynomials.

Step 2: Distribute the Negative Sign

Apply the negative sign from the subtraction to each term inside the second parenthesis. This means changing the signs: \( -(2y^{2} + y + 1) = -2y^{2} - y - 1 \).

Step 3: Combine Like Terms

Rewrite the expression without parentheses: \( 5y^{2} - 3y - 1 - 2y^{2} - y - 1 \). Now combine like terms:- Combine \(5y^{2}\) and \(-2y^{2}\): \(3y^{2}\)- Combine \(-3y\) and \(-y\): \(-4y\)- Combine \(-1\) and \(-1\): \(-2\).

Step 4: Write the Simplified Expression

The simplified result of adding or subtracting the given polynomials is \(3y^{2} - 4y - 2\).

Key Concepts

Distributing Negative Sign

When subtracting polynomials, distributing the negative sign is a crucial initial step. It ensures each term within a polynomial changes its sign. Subtraction can be tricky because of how signs affect each term.Here's a quick guide:

• Identify the polynomial being subtracted. It's enclosed in parentheses and follows the subtraction sign.
• Change the sign of each term within this polynomial. If a term is positive, it becomes negative, and vice versa.

So, if you have \(\left(5y^{2} - 3y - 1\right) - \left(2y^{2} + y + 1\right)\), the negative sign in front of \(2y^{2} + y + 1\) distributes to give \(-2y^{2} - y - 1\). This distribution is important because it keeps the equation balanced as you move to the next steps.

Combining Like Terms

After distributing the negative sign, combining like terms is the next important step. "Like terms" are terms that have the same variables raised to the same power. These can be combined by simply adding or subtracting their coefficients.For example:

• With terms \(5y^{2} - 2y^{2}\), the like terms are \(y^{2}\). Their coefficients are 5 and -2, thus, combine to \(3y^{2}\).
• With \(-3y\) and \(-y\), both share the variable \(y\). Combine their coefficients to get \(-4y\).
• For constant terms \(-1\) and \(-1\), simply add them together to get \(-2\).

By combining these like terms, the expression transforms from a longer list into a simplified, easily readable form with each distinct term type.

Simplifying Polynomials

Once all like terms are combined, the polynomial is simplified. This means presenting the polynomial in its tidiest form, with terms written in descending order of their degree.Let's consider our example:

• You start with the expression: \(5y^{2} - 3y - 1 - 2y^{2} - y - 1\).
• After combining like terms, you should have \(3y^{2} - 4y - 2\).

This is now the simplest form because:

• "Like terms" have been combined.
• The degrees of terms are in descending order: \(y^{2}\), \(y\), constant.
• There are no more operations or reductions possible.

Simplifying polynomials this way makes them easier to understand and work with, especially in further mathematical operations or equations.