Problem 71 - Algebra 1: Concepts and Skills

Question

Evaluate the expression when \(x=-2\) (Lessons \(1.3,2.3,2.5)\). $$ 2 x^{3}+2 x+2 $$

Step-by-Step Solution

Answer

The evaluated expression when \(x = -2\) is -18.

Step 1: Substitute the value of \(x\)

To start solving, one must substitute \(x = -2\) into the given equation \(2x^3 + 2x + 2\). So the expression becomes \(2(-2)^3 + 2(-2) + 2\).

Step 2: Calculation

Now, let's calculate the value: \( -2^3 = -8\). By multiplying 2 with -8, one gets -16. Now, multiplying 2 with -2 gives -4. In the end, we have the equation \(2(-8) + (2*-2) + 2 = -16 - 4 + 2\).

Step 3: Simplify the expression

To find the final solution, one must simplify the obtained expression. So, -16 - 4 + 2 comes out to be -18.

Key Concepts

Substitution

Substitution is a fundamental technique used in algebra to evaluate expressions. Essentially, it involves replacing a variable with a given numerical value. In this context, it allows us to find the actual value of an algebraic expression when specific values are substituted into it.
Let's see how this works with an example: In our exercise, you have the expression \(2x^3 + 2x + 2\), and you're provided that \(x = -2\).

• First, replace every instance of \(x\) in the given expression with \(-2\).
• The expression becomes \(2(-2)^3 + 2(-2) + 2\).
• This approach allows us to transform an algebraic expression into a numerical one that we can further simplify.

By carefully substituting values, we can solve complex expressions step-by-step while maintaining mathematical accuracy.

Simplifying Expressions

Simplifying expressions is a critical step after substituting values into algebraic equations. The process involves combining like terms and performing arithmetic operations to arrive at a simpler form.
Once you've substituted \(x = -2\) into the expression \(2(-2)^3 + 2(-2) + 2\), the next step is to simplify it.

• Begin by calculating each term's value: \((-2)^3\) equals \(-8\).
• Multiply it with 2, resulting in \(-16\).
• Similarly, compute \(2 \times (-2) = -4\).
• Now, your equation becomes \(-16 - 4 + 2\).
• Perform the arithmetic to simplify: First add \(-16\) and \(-4\) to get \(-20\), then add 2 to obtain the final simplified value, \(-18\).

Keep in mind that simplifying expressions not only makes them easier to understand but also helps in arriving at a correct answer effectively.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can involve numbers, operations, and variables. These expressions are foundational in algebra and can range from simple to complex.
For instance, in the exercise \(2x^3 + 2x + 2\), we see a combination of constants and variables. The expression involves:

• Constants, which are the numbers on their own, such as 2 in this equation.
• Variables, which are symbols that can take various values. In our example, \(x\) represents the variable.
• Operators, such as addition and multiplication, which dictate how the numbers and variables interact.

Understanding how these components work together is crucial for solving and interpreting algebraic expressions. They allow us to model real-world problems and find unknown values by manipulating and evaluating these mathematical phrases.