Problem 72 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ x-3 x\left(x^{2}-7 x-1\right) $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression \(x - 3x(x^2 - 7x - 1)\). Answer: \(-3x^3 + 21x^2 - 2x\)

Step 1: Distribute #-3x# inside the parenthesis

To distribute \(-3x\) inside the parenthesis, simply multiply \(-3x\) by each term inside the parenthesis. This will give us: $$ x - 3x(x^2) + 3x(7x) + 3x(-1) $$

Step 2: Perform the multiplications

Now, multiply each term as mentioned in step 1: $$ x - 3x^3 + 21x^2 - 3x $$

Step 3: Combine like terms

Combine the terms with the same exponent of x: $$ -3x^3 + 21x^2 + (x - 3x) $$ Since there is only one cubic term, one quadratic term, and two linear terms, we just need to combine the linear terms: $$ -3x^3 + 21x^2 - 2x $$ So, the simplified algebraic expression is: $$ -3x^3 + 21x^2 - 2x $$

Key Concepts

Distributive Property

The distributive property is a cornerstone of algebra that allows us to simplify expressions and equations by spreading or 'distributing' a multiplication over terms within parentheses. In practical terms, it means that when you have an expression like \( a(b + c) \), you can distribute the \( a \) across the \( b \) and \( c \) to get \( ab + ac \).

For example, let’s look at our exercise problem, where we have to distribute \( -3x \) within the parenthesis \( (x^2 - 7x - 1) \). According to the distributive property, we multiply \( -3x \) by each term inside the parentheses which leads to \( -3x \times x^2 \) for the first term, \( -3x \times -7x \) for the second term, and \( -3x \times -1 \) for the last term. The result of this distribution gives us multiple expressions that we can later combine.

Combining Like Terms

Combining like terms is a process to simplify algebraic expressions or equations. Like terms are terms that have the same variables raised to the same powers. For instance, \( 2x \) and \( 5x \) are like terms because they both have the variable \( x \) with the same exponent. However, \( 2x \) and \( 2x^2 \) are not like terms as the exponents are different.

After distributing, like terms may emerge that can be combined to simplify the expression further. In our exercise, after performing the distribution and multiplication steps, we find terms in the expression \( x, -3x^3, 21x^2, \) and \( -3x \). Then we combine the like terms, which are the linear terms \( x \) and \( -3x \) to get \( -2x \) as the simplified linear component. The other terms, \( -3x^3 \) and \( 21x^2, \) do not have like terms, so they remain as they are in the final simplified expression.

Algebraic Operations

Algebraic operations include addition, subtraction, multiplication, and division, similar to arithmetic operations but involving variables. When we work with algebraic expressions, such as the exercise at hand, we apply these operations following the order of operations which is: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

In our exercise, we initially distribute and multiply, using algebraic multiplication. Subsequently, we simplify by combining like terms, which involves addition or subtraction of coefficients. The goal here is to perform these operations correctly to achieve the most simplified version of the algebraic expression, which in this case is \( -3x^3 + 21x^2 - 2x \).

• Remember to always perform operations within parentheses first.
• Multiplication or division comes next, and here, it is important to handle coefficients and variables with care, especially when dealing with negative signs.
• The last step in simplification will often involve adding or subtracting like terms, thus reducing the expression to its simplest form.