Problem 22 - Algebra 2

Question

Simplify. $$ \left(5 y+3 y^{2}\right)+\left(-8 y-6 y^{2}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-3y^2 - 3y\).

Step 1: Identify like terms

First, we need to identify and group like terms in the expression. The expression is \( (5y + 3y^2) + (-8y - 6y^2) \). The like terms here include linear terms (\( y \)) and quadratic terms (\( y^2 \)).

Step 2: Combine linear terms

Combine the linear terms \( 5y \) and \( -8y \). This can be done by simply adding these coefficients: \( 5y - 8y = -3y \).

Step 3: Combine quadratic terms

Now combine the quadratic terms \( 3y^2 \) and \( -6y^2 \) by adding their coefficients: \( 3y^2 - 6y^2 = -3y^2 \).

Step 4: Write the simplified expression

Now that we've combined like terms, the simplified expression is \( -3y^2 - 3y \).

Key Concepts

Like Terms

In algebra, like terms are terms that have the same variables raised to the same power. Identifying like terms is critical for simplifying algebraic expressions. For example, in the expression \( (5y + 3y^2) + (-8y - 6y^2) \), we have:

• Linear terms, which involve the variable \( y \) raised to the power of 1, such as \( 5y \) and \( -8y \).
• Quadratic terms, which involve the variable \( y \) raised to the power of 2, like \( 3y^2 \) and \( -6y^2 \).

Grouping like terms allows us to combine them more easily. This is like sorting socks in similar pairs before putting them in a drawer. It simplifies the expression, making it easier to work with.

Coefficients

The coefficient in a term is the numerical part that is multiplied by the variable. For the term \( 5y \), the coefficient is 5, while in \( -8y \), it is -8. Coefficients tell us how many times the variable is being multiplied. Let's take a closer look:

• In \( 5y \), the coefficient 5 tells us that \( y \) is multiplied by 5.
• In \( 3y^2 \), the coefficient 3 indicates that \( y^2 \) is multiplied by 3.
• For \( -6y^2 \), the coefficient -6 means \( y^2 \) is multiplied by -6.

When simplifying, we add or subtract these coefficients to combine like terms. It's essential because it allows us to simplify expressions to their neatest form, facilitating further calculations.

Linear Terms

Linear terms are those in an expression where the variable is raised to the first power: \( y \). In our example, the linear terms are \( 5y \) and \( -8y \). Simplifying linear terms involves combining them by manipulating their coefficients:

• Add the coefficients: \( 5 + (-8) = -3 \).
• Then, multiply by \( y \): \( -3y \).

This procedure gives the simplified form of the linear portion of the expression, \( -3y \). Linear terms represent straight-line parts of an expression, hence the name 'linear.' By combining these, we simplify the expression to deal with fewer terms.

Quadratic Terms

Quadratic terms are terms where the variable is squared, such as \( y^2 \). Our expression includes the quadratic terms \( 3y^2 \) and \( -6y^2 \).

• Add the coefficients: \( 3 + (-6) = -3 \).
• Then, multiply by \( y^2 \) to get \( -3y^2 \).

Quadratic terms are significant because they express the curved, parabolic parts of functions. In a polynomial, combining quadratic terms simplifies these curves into a single quadratic term. Recognizing and handling quadratic terms correctly ensures that the expression's complexity does not remain unnecessarily high.