Problem 26 - Algebra A Combined Function

Question

Subtract. $$ \left(-6 y^{2}+3 y-4\right)-\left(9 y^{2}-3 y\right) $$

Step-by-Step Solution

Answer

The solution is \(-15y^2 + 6y - 4\).

Step 1: Identify Like Terms

To subtract the polynomials, first identify like terms from each polynomial. The first polynomial is \(-6y^2 + 3y - 4\), and the second polynomial is \(9y^2 - 3y\). The like terms are \(-6y^2\) and \(9y^2\), and \(3y\) and \(-3y\).

Step 2: Distribute the Negative Sign

Next, distribute the negative sign in front of the second polynomial across each term within the parentheses. This changes the sign of each term:\(9y^2 - 3y\) becomes \(-9y^2 + 3y\).

Step 3: Combine Like Terms

For each pair of like terms, combine them by adding or subtracting:- Combine \(-6y^2\) and \(-9y^2\): \(-6y^2 - 9y^2 = -15y^2\)- Combine \(3y\) and \(3y\): \(3y + 3y = 6y\)The constant term, \(-4\), has no like term in the second polynomial, so it stays the same.

Step 4: Write the Final Result

Combine all the simplified terms to write the final expression:\(-15y^2 + 6y - 4\).

Key Concepts

Like Terms

In polynomial subtraction, **like terms** are terms that have the exact same variables raised to the same power. To subtract polynomials effectively, we first need to spot these like terms. Why? Because only like terms can be combined together, whether through subtraction or addition. For the given polynomials, \(-6y^2 + 3y - 4\) and \(9y^2 - 3y\), we identify like terms as follows:

• Both \(-6y^2\) and \(9y^2\) have the variable \(y^2\).
• The terms \(3y\) and \(-3y\) share the same base, \(y\).

Once like terms are identified, we can effectively proceed with the next steps of polynomial subtraction.

Distributive Property

The **distributive property** is an essential principle that facilitates the subtraction of polynomials, especially when dealing with negative signs. When subtracting one polynomial from another, it helps us remove parentheses and correctly adjust the signs of each term in the subtrahend (the polynomial being subtracted).
In our exercise,
we first encounter the polynomial subtraction in the form of \(( -6y^2 + 3y - 4 ) - ( 9y^2 - 3y )\). To deal with the subtraction, we distribute the minus sign across the terms inside the second set of parentheses

• The positive \(9y^2\) becomes \(-9y^2\)
• The negative \(-3y\) becomes positive \(3y\)

Understanding and applying the distributive property ensures that terms are correctly aligned for subsequent combination.

Combining Like Terms

After identifying like terms and distributing the negative sign, our final task is to **combine like terms**. This means performing addition or subtraction on terms that have the same variables and exponents to simplify the expression.
In the simplified expression, we combine the terms as follows:

• Combine \(-6y^2\) and \(-9y^2\) to get \(-15y^2\).
• Next, combine \(3y\) and \(3y\) to reach \(6y\).
• The term \(-4\) stands alone as a constant, so it remains unchanged.

The result after combining is \(-15y^2 + 6y - 4\). This step effectively tidies the expression, giving us the final outcome of the polynomial subtraction.