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The distributive property is a very useful concept in algebra that helps simplify expressions. It is used to eliminate parentheses by distributing a multiplier outside the parentheses across each term inside them.
For example, in the expression, \( x\left( \frac{5}{6} x-8 \right) + 2\left( \frac{3}{4} x-9 \right) \), we use the distributive property to make it simpler.
Steps to apply the distributive property:

• Identify the terms inside the parentheses.
• Multiply each term inside the parentheses by the term outside.

Starting with \( x \left( \frac{5}{6} x-8 \right) \), we distribute the \( x \):
\( x \times \frac{5}{6}x \right) - x \times 8 \ = \frac{5}{6}x^2 - 8x \).
Next, for \( 2 \left( \frac{3}{4} x-9 \right) \), distribute the 2:
\( 2 \times \frac{3}{4} x - 2 \times 9 \ = \frac{6}{4} x - 18 \ = 1.5 x - 18 \)
Distributing terms helps break down complex expressions into simpler ones.

Combining like terms is the next important step after using the distributive property. Like terms are terms that have the same variables raised to the same power.
For instance, in the expression \( \frac{5}{6} x^2 - 8x + 1.5 x - 18 \), we need to combine the terms containing \( x \).
Steps to combine like terms:

• Identify the terms with the same variable.
• Add or subtract the numerical coefficients of these terms.

Looking at our example, we have the terms \( - 8 x \) and \( 1.5x \). We combine them as follows:
\( - 8 x + 1.5 x = - 6.5 x \).
This gives us a simplified expression: \( \frac{5}{6} x^2 - 6.5 x - 18 \). Combining like terms makes equations simpler and easier to solve or further manipulate.

Polynomials are a central concept in algebra. They are expressions that consist of variables and coefficients, connected by addition, subtraction, and multiplication (but not division by a variable).
A polynomial can have one or more terms, and each term includes a variable raised to a non-negative integer power.
In our example, \( \frac{5}{6} x^2 - 6.5 x - 18 \), this is a polynomial. It has three terms:

• \( \frac{5}{6} x^2 \)
• \( - 6.5 x \)
• \( - 18 \)

These types of expressions are very common in algebra and understanding how to work with them is key.
Key points to remember about polynomials:

• A term consists of a coefficient and a variable.
• Coefficients are the numerical part of the terms.
• Variables are symbols representing numbers.
• The degree of a polynomial is the highest power of the variable.

Polynomials can be added, subtracted, multiplied, and even divided (though special rules apply for division). They are an essential part of algebraic operations and equations.