Problem 52 - Algebra A Combined Function

Question

Simplify each polynomial by combining any like terms. See Examples 13 and 14. $$ -9 x y+7 y-x y-6 y $$

Step-by-Step Solution

Answer

The simplified polynomial is \(-10xy + y\).

Step 1: Identify Like Terms

In the polynomial \(-9xy + 7y - xy - 6y\), we need to identify like terms that can be combined. Like terms are terms that have the same variable components raised to the same powers. In this case, \(-9xy\) and \(-xy\) are like terms, and \(7y\) and \(-6y\) are like terms.

Step 2: Combine Like Terms for 'xy'

Add the coefficients of the like terms with the variables \(xy\):\(-9xy - xy = (-9 - 1)xy = -10xy\).

Step 3: Combine Like Terms for 'y'

Add the coefficients of the like terms with the variables \(y\):\(7y - 6y = (7 - 6)y = 1y\) or simply \(y\).

Step 4: Write the Simplified Expression

Combine the results from Step 2 and Step 3 to obtain the simplified polynomial:\(-10xy + y\).

Key Concepts

Combining Like Terms

To simplify any polynomial expression, the first step is to combine like terms. But what exactly are like terms? Like terms in algebraic expressions have exactly the same variable parts, which means both the variables and their powers are identical. Let's break this down:

• Consider \(-9xy\) and \(-xy\). These are like terms because both have the variables \(x\) and \(y\) with the same power, each raised to the first degree.
• Consider \(7y\) and \(-6y\). These are like terms because both have the variable \(y\) raised to the first degree.

Combining like terms involves performing simple arithmetic on the coefficients associated with these terms. This helps in reducing the polynomial to its simplest form. When you add or subtract coefficients, remember to keep the variable part unchanged, just like grouping apples with apples! With the variables \(xy\), you add \(-9 - 1\) to get \-10xy\, and for \(y\), you calculate \((7-6)\) to end up with \y\.

Polynomial Expressions

Polynomial expressions are mathematical expressions made up of variables and coefficients, structured in terms that include powers of the variables. They are the building blocks of algebra and can range from simple monomials to complex equations with multiple terms. For example:

• The expression \(-9xy + 7y - xy - 6y\) is a polynomial made up of four terms.
• Each term is a combination of a coefficient (number) and variable(s) raised to a power.

Polynomials can be simplified by combining like terms and arranging them in standard form, usually ordered by descending powers of their variables. Once simplified, they are easier to evaluate and use in further mathematical calculations.
Understanding polynomial expressions is crucial for mastering algebra because they form the foundation for more advanced concepts, such as factoring and solving equations. Simplifying these expressions by combining like terms is a powerful tool for making algebraic manipulation more straightforward.

Coefficients in Polynomials

Coefficients are the numerical factors in the terms of a polynomial. They play a critical role in simplifying polynomial expressions by determining how the terms combine. Let's delve into the nature and significance of coefficients:

• In the term \(-9xy\), the coefficient is \(-9\).
• For \(7y\), the coefficient is \7\.
• In \(-xy\), the coefficient is typically \(-1\) because when no number is written, it is implied to be 1.
• The term \(-6y\) has a coefficient of \(-6\).

When simplifying a polynomial, you add or subtract these coefficients for terms that have the same variable components. Understanding coefficients not only aids in algebraic simplification but also paves the way for deeper mathematical topics such as calculus, where coefficients represent rates of change and other phenomena.
By engaging with coefficients, students can see how numbers and variables interact dynamically, creating a pathway to a richer appreciation of algebraic expressions.