Problem 74 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 7 x\left(x^{2}-x+3\right) $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression 7x(x^2 - x + 3). Answer: The simplified algebraic expression is 7x^3 - 7x^2 + 21x.

Step 1: Identify the terms to be distributed

First, we need to identify the terms that need to be distributed. In this case, we have the monomial 7x that needs to be multiplied by each term in the trinomial (x^2 - x + 3).

Step 2: Distribute the monomial to each term in the trinomial

Now, we will distribute the monomial (7x) to each term in the trinomial. This means we have to multiply 7x by each of these three terms: x^2, -x, and 3: 1. 7x * x^2 2. 7x * (-x) 3. 7x * 3

Step 3: Perform the multiplications

Now that we have distributed the monomial, we need to perform the multiplications one by one: 1. 7x * x^2 = 7x^3 (we simply multiply the coefficients and add the exponents for the x terms) 2. 7x * (-x) = -7x^2 (similar to the previous step, multiply coefficients and add exponents) 3. 7x * 3 = 21x (multiply the constant by the coefficient of x and keep the same exponent for x)

Step 4: Combine the results

After performing the multiplications, we can now combine the results to get the simplified algebraic expression: $$ 7x^3 - 7x^2 + 21x $$ So, the simplified algebraic expression for the given problem is: $$ 7x^3 - 7x^2 + 21x $$

Key Concepts

Simplification of Algebraic Expressions

Simplification is the process of reducing an algebraic expression to its simplest form. When simplifying an expression, it's crucial to perform operations in a manner that reduces the expression without changing its value. The goal is to make the expression easier to work with. This means eliminating any unnecessary terms and organizing it into a more manageable format.

In algebra, simplification often involves combining like terms, which are terms that have the same variable parts. Simplification can also mean canceling terms that appear in both the numerator and the denominator of a fraction. However, in the scope of our exercise, we're focusing on simplifying through distribution to deal with less cluttered expressions. The step-by-step procedure involves both distribution and organization, which we will delve into next.

Understanding Distribution in Algebra

Distribution is one of the key concepts when working with algebraic expressions, particularly when you are dealing with polynomials. The distributive property states that for any numbers or variables, \(a(b + c) = ab + ac\). It allows us to expand expressions and simplify them further.

• For the given exercise, distribution involves multiplying the monomial \(7x\) by each term in the trinomial \((x^2 - x + 3)\).
• Executing the distribution results in applying multiplication to each term: \(7x \times x^2\), \(7x \times (-x)\), and \(7x \times 3\).

After distributing the terms, we can simplify the products by multiplying the coefficients and appropriately handling exponents, which ties into our next topic.

Handling Exponents in Algebra

When distributing terms in algebraic expressions, exponents play a significant role. Understanding how to work with them is crucial for simplifying expressions effectively.

The rule of exponents tells us that when multiplying bases, you add their exponents: \(x^a \times x^b = x^{a+b}\). This rule is essential when distributing terms that include variables with exponents, like in our case:

• In \(7x \times x^2 = 7x^{1+2} = 7x^3\).
• Similarly, for \(7x \times (-x) = -7x^{1+1} = -7x^2\).
• The last term simply becomes \(7x \times 3 = 21x\) since multiplying a variable with an exponent of 1 by a constant doesn't change the exponent.

Each step involves straightforward arithmetic but careful attention to applying exponent rules. By handling exponents correctly, we ensure the expression is not only simplified, but also mathematically accurate.