Problem 25 - Elementary and Intermediate Algebra

Question

Simplify each polynomial and write it in descending powers of one variable. $$ -4 a b+4 a b-a b $$

Step-by-Step Solution

Answer

-ab

Step 1: Identify Like Terms

Look at the polynomial \[-4ab + 4ab - ab\]Notice that all terms contain the same variables, \(a\) and \(b\), so they are considered like terms.

Step 2: Combine Like Terms

Add or subtract the coefficients of the like terms. First, combine \(-4ab\) and \(4ab\):\[-4ab + 4ab = 0ab = 0\]Now add the remaining term \(-ab\):\[0 - ab = -ab\]

Step 3: Write the Final Expression

Write the simplified polynomial, which is just one term:\[-ab\]Since it is already in descending order of variables, no further rearrangement is needed.

Key Concepts

Like Terms

Like terms in a polynomial are terms that share the same variables and powers. For instance, in the original expression \(-4ab + 4ab - ab\), all terms consist of the same variables, \(a\) and \(b\). This makes them like terms, which means they can be combined or simplified together. Recognizing like terms is the first and crucial step in simplifying a polynomial. It ensures that you are only combining terms that truly belong together. Like terms are like members of the same team who can work together to achieve a solution, helping to tidy up polynomials neatly for simplification.

Combining Coefficients

Once like terms have been identified, the next step in polynomial simplification is to combine their coefficients. The coefficient is the number in front of the variables. To combine these coefficients, we'll perform basic arithmetic operations such as addition or subtraction. In our initial polynomial expression \(-4ab + 4ab - ab\), here's how you do it:

• Start with the first two terms, \(-4ab\) and \(+4ab\). Their coefficients, \(-4\) and \(+4\), add up to zero, simplifying the expression to \(0\).
• Next, include the last term \(-ab\). Since adding zero does not change the value, the expression becomes \(-ab\).

This process makes the expression much simpler and more concise, helping you easily track the results.

Descending Powers

Organizing a polynomial in descending powers implies arranging terms so that the degrees of their variables decrease from left to right. Although in this exercise there is only one term, it is already in descending order, which is a common requirement when working with polynomial expressions. When you simplify any polynomial, writing it in descending powers can make the expression more clear. It generally follows the format that most polynomial equations are expected to be in, which helps in further mathematical operations. In future expressions with multiple terms, this step involves searching for the highest degree of any term and arranging terms starting from that degree down to the lowest. This order brings clarity to the polynomial's structure, especially as expressions grow more complex.