Problem 79 - Intermediate Algebra

Question

Perform each division. \(\frac{15 x^{2}+9 x-3}{27}\)

Step-by-Step Solution

Answer

\(\frac{5}{9}x^2 + \frac{1}{3}x - \frac{1}{9}\)

Step 1: Understand the Division

The problem requires dividing a polynomial by a constant, which means we need to divide each term of the polynomial separately by the constant.

Step 2: Divide Each Term

Divide each term of the polynomial \(15x^2 + 9x - 3\) by \(27\). This means dividing \(15x^2\), \(9x\), and \(-3\) by \(27\) separately.

Step 3: Simplify Each Fraction

Simplify the fractions resulting from the division: 1. \(\frac{15x^2}{27} = \frac{15}{27} \cdot x^2 = \frac{5}{9}x^2\) (after simplifying \(\frac{15}{27}\) by dividing by their GCD, which is 3)2. \(\frac{9x}{27} = \frac{9}{27} \cdot x = \frac{1}{3}x\) (after simplifying \(\frac{9}{27}\) by dividing by 9)3. \(\frac{-3}{27} = -\frac{3}{27} = -\frac{1}{9}\) (after simplifying \(\frac{-3}{27}\) by dividing by 3)

Step 4: Write the Final Simplified Expression

Compile the simplified terms to write the final answer as the simplified polynomial: \(\frac{5}{9}x^2 + \frac{1}{3}x - \frac{1}{9}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is a key part of many math tasks, including polynomial division. To simplify a fraction, we need to reduce it to its simplest form. This means making the numerator (the top number) and the denominator (the bottom number) as small as possible while keeping their ratio the same.

Here's how it works in practice:

• Find a common factor of the numerator and the denominator.
• Divide both the numerator and the denominator by this common factor.
• Repeat this process until it cannot be reduced any further.

In the equation \(\frac{15x^2}{27}\), the greatest common factor of 15 and 27 is 3. By dividing both by 3, the simplified form is \(\frac{5}{9}\). This process makes calculations easier and results in the simplest form of each polynomial term.

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial concept when simplifying fractions. The GCD between two numbers is the largest number that can divide both without leaving a remainder. This helps in reducing fractions to their simplest forms.

Here's a simple way to find the GCD:

• List all the factors of the first number.
• List the factors of the second number.
• Identify the largest common factor from both lists.

For example, to simplify \(\frac{15}{27}\), list the factors:

• Factors of 15: 1, 3, 5, 15
• Factors of 27: 1, 3, 9, 27
• The GCD is 3.

Dividing both 15 and 27 by 3 gives you \(\frac{5}{9}\), effectively utilizing the GCD to simplify fractions in polynomial division.

Polynomial Terms

Polynomial terms are individual parts of a polynomial expression, separated by plus (+) or minus (-) signs. Each term consists of coefficients and variables raised to a power. For example, in the polynomial \(15x^2 + 9x - 3\), each of these sections represents a term.

To break it down:

• \(15x^2\) is a term where 15 is the coefficient and \x^2\ is the variable with an exponent.
• \(9x\) is a linear term with a variable raised to the power of 1.
• \(-3\) is a constant term, as it does not have a variable part.

Understanding each component helps in performing tasks such as simplification and division, where treating each term appropriately is necessary for accurate calculations.