Problem 53 - Algebra A Combined Function

Question

Simplify each polynomial by combining any like terms. See Examples 13 and 14. $$ 4 x^{2}-6 x y+3 y^{2}-x y $$

Step-by-Step Solution

Answer

The simplified polynomial is \(4x^2 - 7xy + 3y^2\).

Step 1: Identify Like Terms

First, examine the polynomial \( 4x^2 - 6xy + 3y^2 - xy \). Identify like terms based on having the same variables raised to the same powers. Here, \(-6xy\) and \(-xy\) are like terms since they both have the variables 'xy' with no exponents.

Step 2: Combine Like Terms

Add or subtract coefficients of like terms. For the polynomial \(-6xy - xy\), combine to get \(-7xy\). So, the polynomial becomes \(4x^2 - 7xy + 3y^2\).

Step 3: Write the Simplified Polynomial

After combining like terms, write the new simplified polynomial. Therefore, the simplified version of the original polynomial is \(4x^2 - 7xy + 3y^2\).

Key Concepts

Understanding Like Terms

When working with polynomials, one of the most important skills is identifying "like terms." Like terms are terms within an expression that have the same variables raised to the same powers. This means that the coefficients of like terms can be different, but the variable part must be identical.

• For instance, in the polynomial expression \(4x^2 - 6xy + 3y^2 - xy\), like terms are any terms that have the same combination of variables.
• Here, \(-6xy\) and \(-xy\) are like terms because they both share the variables \(x\) and \(y\), with no additional exponents.

Recognizing like terms is crucial for simplifying polynomials because these are the terms we can "combine" to reduce the expression. It makes the expression easier to read and understand.

Combining Like Terms

Once you've identified like terms in a polynomial, the next step is to "combine" them. This process involves adding or subtracting the coefficients (the numbers in front of the variables) of these terms.

• For example, consider \(-6xy\) and \(-xy\). You can rewrite \(-xy\) as \(-1xy\) to see more plainly that you are adding \(-6\) and \(-1\).
• Combining them gives \(-6 - 1 = -7\), so the expression becomes \(-7xy\).

By combining like terms, you simplify the polynomial. This reduces complexity and makes it easier to understand or solve further operations.

Exploring Polynomials

A polynomial is a mathematical expression made up of variables raised to whole number exponents and coefficients. It can include constants, variable terms, and several different kinds of terms:

• A term in a polynomial might look like \(4x^2\), where \(4\) is the coefficient and \(x^2\) is the variable raised to an exponent.
• Polynomials can have multiple terms, including constants like \(3y^2\).

In the example \(4x^2 - 6xy + 3y^2 - xy\), we have:

• One \(x^2\) term, which is \(4x^2\).
• Two \(xy\) terms, which are \(-6xy\) and \(-xy\), making them like terms.
• A single \(y^2\) term, \(3y^2\).

Polynomials are fundamental in algebra, making understanding them essential for further mathematical concepts and operations.