Problem 19 - Elementary and Intermediate Algebra

Question

Simplify each polynomial and write it in descending powers of one variable. $$ \frac{1}{5} x^{2}-\frac{3}{8} x+\frac{2}{3} x^{2}+\frac{1}{4} x $$

Step-by-Step Solution

Answer

\(\frac{13}{15}x^2 - \frac{1}{8}x\)

Step 1: Combine Like Terms

Identify terms that have the same variable and exponent. In this polynomial, the like terms are the ones with \(x^2\) and the ones with \(x\).- Terms with \(x^2\) are \(\frac{1}{5}x^2\) and \(\frac{2}{3}x^2\).- Terms with \(x\) are \(-\frac{3}{8}x\) and \(\frac{1}{4}x\).

Step 2: Simplify \(x^2\) Terms

Add the coefficients of terms containing \(x^2\):\[\frac{1}{5}x^2 + \frac{2}{3}x^2 = \left(\frac{1}{5} + \frac{2}{3}\right)x^2\]To add the fractions, find a common denominator:- The common denominator for 5 and 3 is 15.Convert the fractions:\[\frac{1}{5} = \frac{3}{15}\]\[\frac{2}{3} = \frac{10}{15}\]Now add the fractions:\[\frac{3}{15} + \frac{10}{15} = \frac{13}{15}\]So, the simplified \(x^2\) terms are \(\frac{13}{15}x^2\).

Step 3: Simplify \(x\) Terms

Add the coefficients of terms containing \(x\):\[-\frac{3}{8}x + \frac{1}{4}x = \left(-\frac{3}{8} + \frac{1}{4}\right)x\]Convert the fractions to a common denominator:- The common denominator for 8 and 4 is 8.Convert the fraction:\[\frac{1}{4} = \frac{2}{8}\]Now add the fractions:\[-\frac{3}{8} + \frac{2}{8} = -\frac{1}{8}\]So, the simplified \(x\) terms become \(-\frac{1}{8}x\).

Step 4: Combine and Write in Descending Order

Combine the simplified \(x^2\) and \(x\) terms in descending order of the variable degree:\[\frac{13}{15}x^2 - \frac{1}{8}x\]This is the polynomial simplified and expressed in descending order based on the power of \(x\).

Key Concepts

Combining Like Terms

Combining like terms is an essential step in simplifying polynomials. When simplifying polynomials, we look for terms with the same variable raised to the same power. These are known as "like terms."
Here is how you can combine them effectively:

• Identify terms with the same variable and exponent. In our example, terms with \(x^2\) and terms with \(x\) are like terms.
• Group these like terms together. This helps to focus on them one at a time, making the simplification process clearer.
• Add or subtract the coefficients of these like terms. Only the coefficients change; the variable parts remain unchanged.

For instance, in our exercise, we begin by recognizing \(\frac{1}{5}x^2\) and \(\frac{2}{3}x^2\) as like terms since both contain \(x^2\). Similarly, \(-\frac{3}{8}x\) and \(\frac{1}{4}x\) are like terms because both contain just \(x\). Combining like terms simplifies the expression and reduces it to a more manageable form.

Descending Powers

Descending powers refer to arranging polynomial terms in order, starting with the highest power of the variable down to the lowest. This helps in maintaining a standard form that is easy to read and understand.

• First, identify the highest power of the variable in the polynomial. This will be the leading term.
• Rearrange the polynomial starting with this highest power, followed by terms with lower powers sequentially.
• Ensure all terms are included, even if the coefficient is zero or fraction.

For example, if a polynomial is \(\frac{13}{15}x^2 - \frac{1}{8}x\), the \(x^2\) term comes before the \(x\) term, as \(x^2\) has a higher degree. This clear arrangement makes it easier to compare and simplify or use in further calculations. Using descending powers caters to a uniform method across different math problems.

Fractions in Polynomials

Working with fractions in polynomials can seem complex, but it follows straightforward rules. Here's how to manage them efficiently:

• Fractions appear as coefficients in polynomials. Treat them like any other fraction problems by finding common denominators for addition or subtraction.
• Convert fractions to a common denominator. This is a critical step for adding or subtracting terms.
• Simplify the fractions after adding or subtracting. If possible, reduce them to their simplest form.

In our exercise, terms like \(\frac{1}{5}x^2\) and \(\frac{2}{3}x^2\) require a common denominator to be combined. By converting to a common denominator of 15, we get \(\frac{13}{15}x^2\). Similarly, for \(-\frac{3}{8}x\) and \(\frac{1}{4}x\), a common denominator of 8 allows them to be combined into \(-\frac{1}{8}x\). Handling fractions this way is vital to ensure the polynomial is fully simplified.