Problem 52 - Algebra A Combined Function

Question

Subtract \(\left(5 y^{2}+8 y+2\right)\) from \(\left(7 y^{2}+9 y-8\right)\).

Step-by-Step Solution

Answer

The result of the subtraction is \(2y^2 + y - 10\).

Step 1: Write Down the Expression

To subtract one polynomial from another, start by writing down the two expressions with a subtraction sign in between: \[(7y^2 + 9y - 8) - (5y^2 + 8y + 2)\]

Step 2: Remove Parentheses

Distribute the subtraction sign (or the negative sign) to each term in the second polynomial, changing the signs of those terms. This yields:\[7y^2 + 9y - 8 - 5y^2 - 8y - 2\]

Step 3: Combine Like Terms

Combine like terms by grouping terms with the same power of \(y\):- Square terms: \(7y^2 - 5y^2\)- Linear terms: \(9y - 8y\)- Constant terms: \(-8 - 2\)Now perform the operations:- \(7y^2 - 5y^2 = 2y^2\)- \(9y - 8y = 1y\)- \(-8 - 2 = -10\)

Step 4: Write the Final Expression

Combine the results from Step 3 to form the final expression:\[2y^2 + y - 10\]

Key Concepts

like terms

Recognizing and working with 'like terms' is a fundamental skill in algebra, especially when dealing with polynomials. In a polynomial, like terms are terms that have the same variables raised to the same power. For example, in the exercise provided, the terms involving the variable 'y' raised to the second power, such as \(7y^2\) and \(5y^2\), are considered like terms. Like terms can be combined by adding or subtracting their coefficients.
Understanding like terms allows you to simplify polynomials effectively. When subtracting polynomials, identifying like terms helps you to 'pair up' the corresponding parts of each polynomial so that you can perform subtraction accurately. This is crucial for reducing polynomials to their simplest form.

• Square terms: Look for terms like \(y^2\).
• Linear terms: These involve 'y' without an exponent, like \(9y\) and \(8y\).
• Constant terms: Numbers on their own, such as 2 and -8.

By recognizing and regrouping these terms, the subtraction process becomes much clearer and more manageable.

negative sign distribution

Distributing a negative sign when subtracting polynomials is a vital step that can often cause confusion but is essential for accuracy. When you encounter subtraction between two polynomials, the entire second polynomial is essentially being subtracted from the first.
You achieve this by distributing the negative sign across all terms in the second polynomial. This means you must change the sign of every single term in this polynomial. For instance, in our exercise:

\( (5y^2 + 8y + 2) \) becomes \(-5y^2 - 8y - 2\)

It's like applying a minus sign to each component separately. This distribution ensures that you are accurately subtracting the second polynomial.
If we ignore this crucial step, we risk changing the entire outcome of the problem. By flipping the sign of every term individually, you maintain the structure and value relationships within the equation, leading to a correct solution.

combining polynomials

When we talk about 'combining polynomials,' we mean either adding or subtracting polynomials, which involves carefully aligning and operating on like terms. In the context of subtraction, after applying the negative sign distribution, you must go through the process of simplifying the expression by combining like terms.
This means adding or subtracting the coefficients of like terms:

• For square terms like \(7y^2\) and \(-5y^2\), subtract to get \(2y^2\).
• For linear terms such as \(9y\) and \(-8y\), subtract to get \(y\).
• For constant terms like \(-8\) and \(-2\), combine them to get \(-10\).

Combining polynomials involves meticulous arithmetic computations but always results in a simplified polynomial. It ensures the original expression's value and structure are preserved and correctly recalculated. By methodically simplifying the polynomial, we arrive at the final expression, which, in the exercise, becomes \(2y^2 + y - 10\).
This step confirms the careful subtraction and combination of each term, leading to a precise and streamlined expression.