Problem 24 - Algebra 2

Question

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

Step-by-Step Solution

Answer

\( p(a^2) = 3a^4 - 2a^2 + 5 \)

Step 1: Understanding the Function

We need to substitute the given value into the function. The problem requires us to evaluate the polynomial \( p(x) = 3x^2 - 2x + 5 \) with \( x = a^2 \).

Step 2: Substitute \( x = a^2 \) into \( p(x) \)

Substitute \( a^2 \) for \( x \) in the function \( p(x) \). Therefore, we get: \[ p(a^2) = 3(a^2)^2 - 2(a^2) + 5 \]

Step 3: Simplify \( (a^2)^2 \) to \( a^4 \)

Simplify the expression to deal with \( (a^2)^2 \). Recall that \( (a^m)^n = a^{m\cdot n} \): \[ (a^2)^2 = a^{2\cdot 2} = a^4 \]

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

Substitute back to complete the substitution:\[ p(a^2) = 3a^4 - 2a^2 + 5 \]

Step 5: Collect the Final Expression

Thus, the expression for \( p(a^2) \) is fully expanded and simplified:\[ p(a^2) = 3a^4 - 2a^2 + 5 \]

Key Concepts

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, involving sums of powers of variables. For example, a polynomial function can be written as: - \(p(x) = 3x^2 - 2x + 5\), where \(x\) is the variable. A polynomial is characterized by terms like \(3x^2\), \(-2x\), and \(5\), where each "term" consists of a coefficient (e.g., 3 or -2) and a variable raised to a power (e.g., \(x^2\)). Different powers of the variables (like \(x\) raised to 2) in the terms give each term a different degree.Attributes of polynomial functions include:

• They have a finite number of terms.
• The powers of the variable are non-negative integers.
• They are continuous and smooth curves when graphed.

Understanding these basic properties helps when evaluating polynomial functions for different values.

Function Substitution

Function substitution involves replacing the variable in a polynomial function with another expression or value. In our exercise, we are tasked with evaluating the polynomial \(p(x) = 3x^2 - 2x + 5\) at \(x = a^2\). Here's how it works: - Substitute \(a^2\) for \(x\). This action changes the entire function to focus on the new input: \(p(a^2) = 3(a^2)^2 - 2(a^2) + 5\).Substitution helps us explore the behavior of the polynomial under different conditions. It transforms the original polynomial's output based on new inputs.This process requires care and precision, as mistakes can alter the entire outcome. It's crucial to replace every occurrence of the variable uniformly.

Simplification

Simplification is the process of reducing complex expressions to their simplest form. In our context, we simplify \(3(a^2)^2 - 2(a^2) + 5\) by applying mathematical operations like multiplication and subtraction.Simplification makes expressions easier to interpret and solve. Here's the breakdown in our example:

• Transform \( (a^2)^2 \) to \(a^4\)
• Multiply \(3 imes a^4 = 3a^4\)
• Multiply \(-2 imes a^2 = -2a^2\)
• Combine results into one simplified expression: \(3a^4 - 2a^2 + 5\)

By eliminating redundant steps and merging terms, simplification makes computations cleaner and more understandable.

Exponent Laws

Exponent laws are rules that simplify calculations involving powers of numbers or variables. Understanding these rules is crucial when working with expressions like \(3(a^2)^2\). Let's explore the key laws: - Power of a Power: For \((a^m)^n = a^{m\cdot n}\), simplify by multiplying the exponents. In our example, \((a^2)^2 = a^{2\cdot 2} = a^4\).These laws help us handle expressions where one power is raised to another, ensuring accurate and efficient simplification.Each law provides a method to break down complex expressions into manageable parts, leading to faster and error-free calculations.