Problem 25 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The value is \(12a^6 - 4a^3 + 5\).

Step 1: Understand the Function

The given function is \(p(x) = 3x^2 - 2x + 5\). We are supposed to evaluate this function at \(x = 2a^3\).

Step 2: Substitute \(x = 2a^3\)

Replace \(x\) with \(2a^3\) in the polynomial function \(p(x)\). This gives us the expression: \(p(2a^3) = 3(2a^3)^2 - 2(2a^3) + 5\).

Step 3: Simplify \((2a^3)^2\)

Calculate \((2a^3)^2\). This is \(4a^6\), because \((2a^3)^2 = 2^2\cdot(a^3)^2 = 4a^{6}\).

Step 4: Substitute \(4a^6\) into the Expression

Substitute \(4a^6\) into the expression obtained in Step 2: \(p(2a^3) = 3(4a^6) - 2(2a^3) + 5\).

Step 5: Simplify the Polynomial Expression

Compute the terms in the expression: - \(3(4a^6) = 12a^6\)- \(-2(2a^3) = -4a^3\)Now, the expression is \(12a^6 - 4a^3 + 5\).

Step 6: Combine All Terms

The final expression simplifies to \(12a^6 - 4a^3 + 5\). This is the value of \(p(2a^3)\).

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical phrases that involve a sum of powers of variables with their coefficients. These can include various terms, such as linear terms (like \(x\)), quadratic terms (like \(x^2\)), and even higher-degree terms. Each term in a polynomial consists of coefficients and variables raised to whole number powers. In general, a polynomial expression in one variable, \(x\), takes the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n, a_{n-1}, ... , a_0\) are constants.

In the exercise, the polynomial given was \(p(x) = 3x^2 - 2x + 5\). It is a second-degree polynomial because the highest power of \(x\) is 2. Understanding polynomial expressions helps in breaking down the individual components of an expression for evaluation or simplification.

Variable Substitution

Variable substitution is a technique used to evaluate expressions by replacing variables with specific values or expressions. It's a crucial step in solving many algebraic problems, especially when evaluating polynomial functions.

In the provided exercise, the function \(p(x)\) needed to be evaluated at \(x = 2a^3\). This involves substituting \(2a^3\) into each instance of \(x\) in the polynomial expression. The resulting expression becomes \(p(2a^3) = 3(2a^3)^2 - 2(2a^3) + 5\).

Substitution simplifies analysis as it converts a general expression into a specific one that can be further manipulated. It's important to perform this step carefully to ensure that each substitution is applied consistently across the expression.

Algebraic Simplification

Algebraic simplification involves reducing an expression to its simplest form. It requires performing operations such as distribution, combining like terms, and reducing coefficients and powers wherever possible.

In the exercise, once substitution was complete, simplifying the expression \((2a^3)^2\) produced \(4a^6\), because \((2a^3)^2 = (2)^2(a^3)^2 = 4a^6\). After this, the expression \(3(4a^6) - 2(2a^3) + 5\) needed further simplification. This includes evaluating each term:

• \(3(4a^6) = 12a^6\)
• \(-2(2a^3) = -4a^3\)
• The constant term \(+5\) remains unchanged.

The final simplified expression is \(12a^6 - 4a^3 + 5\). Simplification makes complex expressions easier to work with and understand by breaking them down into smaller, more manageable pieces.