Problem 46 - Algebra 2

Question

If \(p(x)=2 x^{2}-5 x+4\) and \(r(x)=3 x^{3}-x^{2}-2,\) find each value. $$ r(2 a) $$

Step-by-Step Solution

Answer

\(r(2a) = 24a^3 - 4a^2 - 2.\)

Step 1: Understand the problem statement

We have been given the polynomial function \( r(x) = 3x^3 - x^2 - 2 \) and need to find the value of \( r(2a) \). This involves substituting \( 2a \) into the polynomial function \( r(x) \) and simplifying the expression.

Step 2: Substitute \( 2a \) into the polynomial

Replace every instance of \( x \) in the function \( r(x) = 3x^3 - x^2 - 2 \) with \( 2a \). This gives us the expression: \[ r(2a) = 3(2a)^3 - (2a)^2 - 2.\]

Step 3: Simplify the expression

Calculate \( (2a)^3 \) which is \( 8a^3 \), and multiply by 3 to get \( 24a^3 \). Next, calculate \( (2a)^2 \) which is \( 4a^2 \). Substitute these back into the expression to get:\[ r(2a) = 24a^3 - 4a^2 - 2.\]

Step 4: Present the simplified expression

The final simplified expression for \( r(2a) \) is: \[ r(2a) = 24a^3 - 4a^2 - 2.\]

Key Concepts

Polynomial Evaluation

Polynomial evaluation is a process where we calculate the value of a polynomial function at a given point or expression.
Polynomials are mathematical expressions involving a sum of powers of variables multiplied by coefficients.
For example, in the polynomial \( r(x) = 3x^3 - x^2 - 2 \), the goal of evaluation is to compute what the value of \( r(x) \) is when \( x \) takes on a specific value.

• This evaluation can be done by directly substituting the given value or expression into the polynomial.
• It simplifies calculations, especially when using polynomials to model real-world scenarios and obtain specific data points.

Here, for \( r(2a) \), we replace every instance of \( x \) in \( r(x) \) with \( 2a \), which initiates the evaluation.

Substitution Method

The substitution method in mathematics involves replacing a variable with a given value or expression.
This method is crucial for evaluating polynomials, such as substituting \( x \) with \( 2a \) in the polynomial \( r(x) = 3x^3 - x^2 - 2 \).

• Identify the variable to substitute, which is \( x \) in this case.
• Replace each occurrence of \( x \) in the polynomial with the expression \( 2a \).

By doing so, we transform the original polynomial \( r(x) \) into a simpler expression that incorporates the new terms from substitution, making it ready for simplification.

Simplification in Algebra

Simplification in algebra involves reducing expressions into their simplest form.
After substituting \( 2a \) into \( r(x) \), the expression becomes \( r(2a) = 3(2a)^3 - (2a)^2 - 2 \).

• First, compute powers like \( (2a)^3 \) which results in \( 8a^3 \), and then multiply by 3 to get \( 24a^3 \).
• Similarly, compute \( (2a)^2 \) giving \( 4a^2 \).
• Replace these back into the polynomial to obtain \( 24a^3 - 4a^2 - 2 \).

The final expression \( r(2a) = 24a^3 - 4a^2 - 2 \) represents the simplified form, which is now free of unnecessary complexity and ready for interpretation or further use.