Problem 20 - Algebra A Combined Function

Question

Subtract. $$ \left(-7 y^{2}+5\right)-\left(-8 y^{2}+12\right) $$

Step-by-Step Solution

Answer

The solution is \(y^2 - 7\).

Step 1: Distribute the Negative Sign

The problem requires subtracting a polynomial from another. First, distribute the negative sign across the second polynomial inside the brackets: \ \(-7y^2 + 5 - (-8y^2 + 12) = -7y^2 + 5 + 8y^2 - 12\). This step involves changing the signs of the terms inside the second set of parentheses.

Step 2: Combine Like Terms

Now, combine the like terms by adding the coefficients of \(y^2\) terms together and the constant terms together. The expression becomes: \ \((-7y^2 + 8y^2) + (5 - 12)\).

Step 3: Simplify the Expression

Calculate the results for each combined group of terms: \ \(-7y^2 + 8y^2 = 1y^2\) and \(5 - 12 = -7\). This simplifies the expression to \(y^2 - 7\).

Key Concepts

Distributive Property

When working with polynomial subtraction, the distributive property becomes a very useful tool. This property allows us to simplify expressions by distributing multipliers across terms inside parentheses:

• If you have to subtract a group of terms, apply the negative sign to each term within the brackets.
• This changes every term to its opposite.

For instance, in our original problem, the expression \(-(-8y^2 + 12)\) requires the negative sign to be distributed over the terms inside the parentheses. Thus, \(-8y^2\) becomes \(+8y^2\), and \(+12\) becomes \(-12\).
This transformation is essential because it sets the stage for combining like terms, which is the next step in simplifying a polynomial expression.

Combining Like Terms

Once you have distributed any negative signs or multipliers, it is time to combine like terms, a key step in simplifying polynomial expressions. "Like terms" are terms that have the same variable raised to the same power. Here’s what you need to do:

• Identify terms with the same variables and exponents.
• Add or subtract their coefficients accordingly.

In our example, we deal with \(-7y^2 + 8y^2\) and combine them since they are like terms. Similarly, the constant terms \(+5\) and \(-12\) are also like terms that need to be combined:

• For \(-7y^2 + 8y^2\), the coefficients are combined to yield \(+1y^2\) or simply \y^2\.
• For \(+5 - 12\), the result is \-7\.

By combining the like terms, the expression is further simplified and closer to its solution.

Simplifying Expressions

The final goal in polynomial subtraction is to simplify the expression completely. Once you've combined like terms, you'll express the polynomial in its simplest form. Here’s what entails this step:

• Calculate the results of combined terms.
• Write down the final expression without unnecessary parts.

From our problem, after combining the like terms, we calculated:

• \(-7y^2 + 8y^2 = y^2\)
• \(+5 - 12 = -7\)

Thus, the simplified expression is \y^2 - 7\. This expression is as simple as it can get since there are no other like terms to combine or further reductions possible. Understanding each step from distributing negatives to combining like terms is crucial for mastering polynomial subtraction.