Problem 56 - Algebra A Combined Function

Question

Subtract \(\left(y^{2}-9\right)\) from the sum of \(\left(3 y^{2}+y+4\right)\) and \(\left(2 y^{2}-6 y-10\right)\)

Step-by-Step Solution

Answer

The final result is \(4y^2 - 5y + 3\).

Step 1: Find the sum of the first two expressions.

The problem requires finding the sum of the expressions \(3y^2 + y + 4\) and \(2y^2 - 6y - 10\). To do this, combine like terms:\[3y^2 + y + 4 + 2y^2 - 6y - 10\]Combine the \(y^2\) terms: \(3y^2 + 2y^2 = 5y^2\).Combine the \(y\) terms: \(y - 6y = -5y\).Combine the constant terms: \(4 - 10 = -6\).Thus, the sum is \(5y^2 - 5y - 6\).

Step 2: Subtract the third expression

Next, subtract the expression \(y^2 - 9\) from the sum found in Step 1, which is \(5y^2 - 5y - 6\).First, distribute the negative sign to the terms in the expression \(y^2 - 9\):\(- (y^2 - 9) = -y^2 + 9\).Now subtract:\[5y^2 - 5y - 6 - y^2 + 9\]Combine like terms:\(5y^2 - y^2 = 4y^2\),\(-5y\) (there are no other \(y\) terms),\(-6 + 9 = 3\).Thus, the result of the subtraction is \(4y^2 - 5y + 3\).

Key Concepts

Polynomial Addition

Polynomial addition is a fundamental concept in algebra that involves combining like terms from two or more polynomial expressions. To add polynomials, you need to:

• Identify like terms: these are terms that have the same variable raised to the same power. For instance, in the expression \(3y^2 + y + 4\) and \(2y^2 - 6y - 10\), \(3y^2\) and \(2y^2\) are like terms because they both contain \(y^2\).
• Combine these like terms by adding or subtracting the coefficients: In our example, \(3y^2 + 2y^2 = 5y^2\).
• Continue this process for all different powers of the variable present in the polynomials: Combine \(y\) terms to get \(y - 6y = -5y\), and combine constant terms to \(4 - 10 = -6\).

By following these steps, you find the sum of the polynomials: \(5y^2 - 5y - 6\). The beauty of polynomial addition is its simplicity once you've mastered combining like terms.

Polynomial Subtraction

Subtracting polynomials extends the process of addition and involves a similar set of skills. The key difference is that every term of the polynomial being subtracted must have its sign changed before performing any further arithmetic. Let's break it down:

• Firstly, you distribute the negative sign across the terms of the polynomial you'll subtract. For example, when subtracting \(y^2 - 9\) from \(5y^2 - 5y - 6\), we first distribute to get \(-y^2 + 9\).
• Then, align similar terms and change their signs if needed: bring \(5y^2\) and \(-y^2\) together, \(-5y\) stands alone, and combine \(-6 + 9\).
• Apply the same rule of combining like terms as in addition. Here, \(5y^2 - y^2 = 4y^2\) and \(-6 + 9 = 3\).

The final expression is \(4y^2 - 5y + 3\). Careful management of signs is crucial in polynomial subtraction to ensure accuracy.

Combining Like Terms

Combining like terms is perhaps the most crucial part of manipulating algebraic expressions effectively. It simply refers to the process of adding or subtracting terms that have identical variable parts:

• Terms such as \(3y^2\) and \(2y^2\) are like terms because they both have \(y^2\).
• When combining, you focus on the coefficients (the numerical part of the terms): if you add \(3y^2\) and \(2y^2\), you add the numbers \(3\) and \(2\) to get \(5\), which gives you \(5y^2\).
• Remember that only the coefficients change when you combine like terms; the variables and their exponents stay the same.
• Combining constants is also a part of this process, just like in basic arithmetic (e.g., \(4 - 10 = -6\)).

By combining like terms, polynomials are simplified, making it easier to work with or further manipulate the expressions in various types of algebraic operations.

Distributive Property

The distributive property is a core algebraic principle that helps simplify expressions, especially when dealing with subtraction in algebraic expressions. It involves multiplying a term across a sum or difference within parentheses:

• The key idea is that everything inside the parentheses is multiplied by the factor outside. For example, when subtracting \((y^2 - 9)\), you distribute the negative sign to get \(-y^2 + 9\).
• This rule is also useful when expanding expressions like \(a(b + c)\) to \(ab + ac\), ensuring all terms are combined correctly.
• In more complex situations, pay attention to maintaining signs through distribution. Mistakes here often lead to errors in combining terms later.

Overall, the distributive property simplifies managing more complex expressions, aiding in precise and efficient algebraic manipulations.