Problem 7 - Algebra A Combined Function

Question

$$ \left(-3 y^{2}-4 y\right)+\left(2 y^{2}+y-1\right) $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(-y^2 - 3y - 1\).

Step 1: Distribute the Negative Sign

The expression we are given is \(-3y^2 - 4y + 2y^2 + y - 1\). Notice that the terms inside the parentheses after the minus sign are being subtracted. Distribute the subtraction (negative sign) as follows: \(-3y^2 - 4y + (2y^2 + y - 1)\). This becomes \(-3y^2 - 4y + 2y^2 + y - 1\).

Step 2: Combine Like Terms

Look for terms in the expression that are similar, which means they have the same variable raised to the same power. We will group them by \(y^2\) and \(y\) and the constant terms:- For the \(y^2\) terms: \(-3y^2 + 2y^2\) yields \(-y^2\).- For the \(y\) terms: \(-4y + y\) results in \(-3y\).- The constant term is \(-1\).So, the combined expression is \(-y^2 - 3y - 1\).

Key Concepts

Distributing Negative

When dealing with polynomial addition specifically, pay close attention to any negative signs before parentheses. A common first step is to distribute a negative sign properly across terms inside these parentheses. This means every term inside changes its sign. For example, when you have

• \((-3y^2 - 4y) + (2y^2 + y - 1)\)

Traditionally, the negative sign in front of a group of terms indicates that all terms within the parentheses will have their signs flipped. Our task would be to change all the

• positive terms inside the parentheses to negatives,
• and vice versa.

So in this expression, the terms remain with the same signs because they were already simplified outside parentheses.Sometimes the expressions are initially grouped in other formats where distribution needs to be more apparent. Always check expressions next to negative signs to make sure they are properly distributed.

Combining Like Terms

Combining like terms is an essential step in polynomial operations. It implies gathering and simplifying terms with identical variable parts (same variable and exponent). Take the terms of the expression, search for like terms, and simplify them as a unit.In our exercise, we simplify as follows:

• First, look at the terms with the power of 2: \((-3y^2) + (2y^2)\) results in \(-y^2\).
• Next, gather the terms with the simplest power, the linear terms: \((-4y) + (y)\) becomes \(-3y\).
• Finally, constant terms remain unchanged unless further instructions are provided, here it's simply \(-1\).

Remember, the coefficients (numbers in front of terms) are added or subtracted, but the variable part stays the same. This will result in a simplified polynomial that expresses the same values as the original one, but in a more concise form.

Simplifying Expressions

Simplifying expressions aims to rewrite a polynomial in the fewest terms possible, keeping it neat and efficient while maintaining equality to the original expression. This often involves repeatedly combining like terms until no further simplifications can be made. In our given polynomial, after distributing and combining like terms, we end up with:

• \(-y^2 - 3y - 1\)

This process involves looking for potential reductions or cancellations within an algebraic expression. It helps in reducing complexity without altering the underlying mathematical properties. As a student, focusing on identifying terms that can be simplified helps enormously in reducing errors and finding quick paths to the simplest form without unnecessary calculations.