Problem 49 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ x(x+6) $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is $$x^2 + 6x$$.

Step 1: Distribute x

To distribute x to both terms inside the parenthesis, we multiply x by each term. This gives us: $$ x*x + x*6 $$

Step 2: Simplify Terms

Now, we will simplify the obtained expression: $$ x^2 + 6x $$ As there are no like terms to combine, the simplified expression is: $$ x^2 + 6x $$

Key Concepts

Polynomials

Polynomials are fundamental expressions in algebra that consist of variables and coefficients. They involve operations of addition, subtraction, multiplication, and non-negative integer exponents. A simple yet classic example of a polynomial is an expression like \( x + 6 \) or more complex ones such as \( x^2 + 6x + 9 \).

A polynomial can be classified by the number of terms it contains:

• Monomial: A polynomial with one term, such as \( 5x \).
• Binomial: A polynomial with two terms, like \( x + 6 \).
• Trinomial: A polynomial with three terms, such as \( x^2 - 3x + 12 \).

Understanding polynomials is crucial as they form the building blocks of more complex algebraic structures. They are used to represent equations and functions, making them essential for solving mathematical problems.

Distribution Method

The distribution method is a useful technique in algebra, especially when working with polynomials and parentheses. It involves distributing, or multiplying, a single term by each term inside a set of parentheses. This method helps in expanding expressions and simplifying calculations.

For example, in the expression \( x(x+6) \), we distribute the \( x \) by multiplying it with each term inside the parenthesis:

• First, multiply \( x \) with \( x \), resulting in \( x^2 \).
• Next, multiply \( x \) with \( 6 \), giving \( 6x \).

Combining these results, you get \( x^2 + 6x \). This process is essential for expanding expressions and preparing them for further operations such as adding or subtracting like terms.

Like Terms

Like terms are terms in an algebraic expression that have the same variable raised to the same power. They can be combined to simplify expressions, making calculations more straightforward.

When dealing with expressions like \( x^2 + 6x \), it is important to identify terms that are 'like' each other. In this case, \( x^2 \) and \( 6x \) are not like terms because they are raised to different powers (one is squared, the other is not). This means they cannot be combined further.

However, if you had an expression such as \( 3x + 6x \), both terms are 'like terms' because they each have the variable \( x \) raised to the same power (1 in this case). These can be combined to make \( 9x \). Recognizing and combining like terms is a key skill in algebra that allows for simplification and better organization of expressions.