Problem 56 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 9 x(x-3) $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is $9x^2 - 27x$.

Step 1: Apply the distributive property

Multiply the 9 by both terms inside the parentheses: $$ 9x(x-3) = 9x\cdot x - 9x\cdot 3 $$

Step 2: Perform the multiplication

Multiply the terms: $$ 9x\cdot x - 9x\cdot 3 = 9x^2 - 27x $$

Step 3: Combine like terms, if any

In our expression, there are no like terms to combine. So, the final simplified expression is: $$ 9x^2 - 27x $$

Key Concepts

Distributive Property

The Distributive Property is a fundamental algebraic principle that plays a crucial role in simplifying expressions. It's a handy tool that allows you to multiply a single term by two or more terms within a set of parentheses. This property states that if you have a product, say, \( a(b + c) \), you can "distribute" the \( a \) across the terms inside the parentheses.

• Mathematically, it's expressed as \( a(b + c) = ab + ac \).
• For subtraction, the property works similarly: \( a(b - c) = ab - ac \).

In the exercise \( 9x(x-3) \), the distributive property is used to multiply \( 9x \) with both \( x \) and \(-3\). First, \( 9x \cdot x \) gives \( 9x^2 \), and second, \( 9x \cdot (-3) \) results in \(-27x\). Thus, after applying the distributive property, the expression becomes \( 9x^2 - 27x \). This method is efficient in handling expressions with parentheses and combining different terms.

Like Terms

In algebra, "like terms" are terms that contain the same variables raised to the same powers, though they may have different coefficients. Recognizing like terms is essential for simplifying algebraic expressions.

• For example, \( 3x \) and \( 7x \) are like terms because they both contain the variable \( x \). However, \( 3x \) and \( 3x^2 \) are not like terms since the exponent on \( x \) is different.
• Combining like terms simply involves adding or subtracting their coefficients.

In the context of our exercise, \( 9x^2 - 27x \), the terms \( 9x^2 \) and \(-27x\) are not like terms since the variables have different powers (\( x^2 \) and \( x^1 \), respectively). Therefore, they cannot be combined any further, which is why the expression is already simplified.

Multiplication of Polynomials

Multiplying polynomials involves distributing each term in the first polynomial to each term in the second one. This requires careful application of the distributive property and sometimes combining like terms is necessary.

• For a simple multiplication such as \( (a+b)(c+d) \), every term in \( a+b \) must be multiplied by every term in \( c+d \).
• In broader situations like \( 9x(x-3) \), a monomial is multiplied by a binomial. You apply the distributive property to each term.

Here's how it appeared in our exercise: Multiply \( 9x \) by \( x \) to get \( 9x^2 \), and \( 9x \) by \(-3\) to get \(-27x\). Thus, the result after multiplying is \( 9x^2 - 27x \). Multiplying polynomials is a foundational skill in algebra that prepares you for tackling more complex expressions and equations.