Problem 56 - Algebra A Combined Function

Question

Simplify each polynomial by combining any like terms. See Examples 13 and 14. $$ 17 a^{2} b-16 a b^{2}+3 a^{3}+4 b a^{3}-b^{2} a $$

Step-by-Step Solution

Answer

The simplified polynomial is \(17a^2b + 7a^3 - 17ab^2\).

Step 1: Identify Like Terms

First, we need to look for terms with the same variables raised to the same power. These are like terms. In our expression, the terms are \(17a^2b\), \(-16ab^2\), \(3a^3\), \(4ba^3\), and \(-b^2a\). Notice that \(3a^3\) and \(4ba^3\) are like terms because they both contain \(a^3\). Also, the terms \(-16ab^2\) and \(-b^2a\) are like, as they both contain \(b^2a\).

Step 2: Combine Like Terms

Now, combine the like terms identified. For \(3a^3\) and \(4ba^3\), add the coefficients: \(3 + 4 = 7\), giving \(7a^3\). For \(-16ab^2\) and \(-b^2a\), the coefficients are \(-16\) and \(-1\), so add them: \(-16 + (-1) = -17\), giving \(-17ab^2\).

Step 3: Write the Simplified Polynomial

After combining the like terms, rewrite the polynomial. The simplified polynomial is: \[17a^2b + 7a^3 - 17ab^2\]

Key Concepts

Like Terms

In algebra, understanding the concept of like terms makes simplifying polynomials much easier. Like terms are terms that contain the same variables, with each variable raised to the same power. This means you can only combine terms if they have exactly the same variable components.
For example:

• Terms like \( 3a^3 \) and \( 4ba^3 \) are considered like terms because both of them have the variable \( a \) raised to the third power.
• On the other hand, \( 2x \) and \( 2x^2 \) are not like terms. The presence of \( x \) in both is deceptive because the exponent, or power, is different.

Recognizing these like terms in a polynomial helps in simplifying it, leading to an easier and more streamlined expression.

Combining Like Terms

Once you have identified the like terms, the next step is combining them. This means you'll add or subtract their coefficients, which are the numerical factors of the terms.
Let's look at our earlier example:

• If you have the terms \( 3a^3 \) and \( 4ba^3 \), the like terms both have \( a^3 \). So you add the coefficients 3 and 4, resulting in \( 7a^3 \).
• Similarly, for terms like \( -16ab^2 \) and \( -b^2a \), we combine the coefficients \(-16\) and \(-1\) to get \(-17ab^2\).

This process simplifies the polynomial by consolidating similar parts into a single term, making the whole expression easier to understand and work with.

Coefficients

Coefficients are crucial in the operation of combining like terms. The coefficient is the numerical part of a term, sitting in front of the variables. It's what you add or subtract when you combine like terms.
For instance:

• In the polynomial term \( 3a^3 \), the number 3 is the coefficient.
• In the term \( -16ab^2 \), \(-16\) is the coefficient.

Coefficients are numbers that indicate how many times the term is counted. When simplifying, only the coefficients change; the variable parts stay the same if you've correctly combined like terms. This makes understanding coefficients a stepping stone to mastering polynomial simplification.

Polynomials in Algebra

Polynomials are expressions composed of variables and coefficients, structured in a manner that sets algebra apart from basic arithmetic. They are incredibly versatile and form the basis of many algebraic operations. Each segment of a polynomial, known as a "term," can vary in complexity.A polynomial might look like \( ax^n + bx^{n-1} + \, ... \, + k \), where:

• \( a, b, \), and \( k \) represent coefficients
• \( x \) stands for the variables involved
• \( n \) is a non-negative integer determining the degree of the polynomial

Polynomials can have various operations applied to them, including addition, subtraction, and even multiplication and division. Simplification of polynomials involves turning a complex expression into a simpler or more standard form by combining like terms and using basic algebra rules. Understanding polynomials is fundamental in algebra, setting the stage for solving equations and modeling real-world scenarios.