Problem 22 - Algebra 2

Question

If \(p(x)=3 x^{2}-2 x+5\) and \(r(x)=x^{3}+x+1,\) find each value. \(r(3 a)\)

Step-by-Step Solution

Answer

\(r(3a) = 27a^3 + 3a + 1\)

Step 1: Understand the Function

The function given is \(r(x) = x^3 + x + 1\). To find \(r(3a)\), we need to substitute \(3a\) into each occurrence of \(x\) in the function \(r(x)\).

Step 2: Substitute \(3a\) into the Function

Replace \(x\) in the function \(r(x) = x^3 + x + 1\) with \(3a\). This gives us the expression: \(r(3a) = (3a)^3 + 3a + 1\).

Step 3: Calculate \((3a)^3\)

Calculate \((3a)^3\). Recall that \((3a)^3 = 3^3 \cdot a^3 = 27a^3\).

Step 4: Simplify the Expression

Substitute the calculated value back into the expression to get \(r(3a) = 27a^3 + 3a + 1\).

Step 5: Write the Final Expression

The expression for \(r(3a)\) is fully simplified as \(27a^3 + 3a + 1\).

Key Concepts

Function Evaluation

Evaluating a function is a core concept in mathematics. When you evaluate a function, you're essentially finding the value of the function for a specific input. In our example, we are working with the function \(r(x) = x^3 + x + 1\). Evaluating \(r(x)\) means calculating its value for a particular value of \(x\). In function notation, this process is denoted as \(r(3a)\), meaning we want to find the value of \(r\) when \(x\) is replaced by \(3a\).
This step is essential in understanding how a function behaves with different inputs. In real-world scenarios, functions model everything from how a car accelerates to how populations grow. Each different input can give us important insights into specific conditions or predictions.

Substitution in Polynomials

Substitution is a key technique used to evaluate polynomials, which are mathematical expressions involving sums of powers of variables. In this exercise, we need to substitute \(3a\) into the polynomial \(r(x) = x^3 + x + 1\). This means wherever you see \(x\) in the polynomial, you replace it with \(3a\).
For example, in our function, substituting \(3a\) for \(x\) turns each \(x\) into \(3a\). Thus, our function becomes \((3a)^3 + 3a + 1\).

• Start with the original function \(r(x)\).
• Identify all instances of the variable \(x\).
• Replace each \(x\) with the new argument \(3a\).

This substitution will allow us to move forward to the next step, which is simplifying the expression.

Simplifying Expressions

Once substitution is complete, simplifying the expression is the next logical step. Simplifying polynomials involves reducing the expression to its simplest form, combining like terms, and calculating powers. In our example, after hopping from \((3a)^3 + 3a + 1\), we calculated \((3a)^3\) to get \(27a^3\).

• Simplify each component separately, for instance, \((3a)^3 = 3^3 \cdot a^3 = 27a^3\).
• Add up all parts of the polynomial that remain to form the final expression.

For the task at hand, the simplified form of \(r(3a)\) is \(27a^3 + 3a + 1\). Simplifying helps provide the clearest and most useful form of a result, important for both academic purposes and practical problem-solving.