Problem 48 - Algebra 2

Question

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

Step-by-Step Solution

Answer

\( p(2a^2) = 8a^4 - 10a^2 + 4 \).

Step 1: Identify the Given Function

We are given the polynomial function \( p(x) = 2x^2 - 5x + 4 \). Our task is to evaluate \( p(2a^2) \).

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

Replace every occurrence of \( x \) in the function \( p(x) = 2x^2 - 5x + 4 \) with \( 2a^2 \). This gives us the expression \( p(2a^2) = 2(2a^2)^2 - 5(2a^2) + 4 \).

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

Simplify \((2a^2)^2\) by calculating the square: \((2a^2)^2 = 4a^4\).

Step 4: Substitute and Simplify

Substitute \(4a^4\) back into the expression: \( p(2a^2) = 2(4a^4) - 5(2a^2) + 4 \). This simplifies to \( 8a^4 - 10a^2 + 4 \).

Key Concepts

Polynomial Substitution

Polynomial substitution is a fundamental technique in algebra that helps you determine the value of a polynomial function for a given input. This process involves replacing the variable in the polynomial with a specific value or expression. Let's explore this concept with the polynomial given in the exercise: \( p(x) = 2x^2 - 5x + 4 \).

When you're tasked with finding \( p(2a^2) \), you're engaging in polynomial substitution. It involves taking the input value, which in this case is \( 2a^2 \), and substituting it for every occurrence of \( x \) in the polynomial. Here's how it works:

• Start with the original polynomial function.
• Substitute \( x \) with \( 2a^2 \) in the expression.
• This gives: \( p(2a^2) = 2(2a^2)^2 - 5(2a^2) + 4 \).

After substitution, the next step is to simplify the expression by performing the mathematical operations, specifically the calculation of squares and combining like terms.

Polynomial Evaluation

Polynomial evaluation follows after substitution, where we simplify and solve to find a specific value. This step requires a strong grip on evaluating powers and performing algebraic operations. Continuing with the exercise example:

You first calculate the square of the expression that replaces \( x \). For \( (2a^2)^2 \), you multiply the coefficients and variables:

• \( (2a^2)^2 = 2^2 \times (a^2)^2 = 4a^4 \)

With this result, you substitute it back into the polynomial.

Next, let's break down the polynomial evaluation step-by-step:

• Replace each squared term and variable: \( 2(4a^4) - 5(2a^2) + 4 \).
• Multiply the coefficients: \( 8a^4 - 10a^2 + 4 \).
• Combine these calculated terms to get the simplified expression.

The final expression \( 8a^4 - 10a^2 + 4 \) is the evaluation of the polynomial for \( x = 2a^2 \), offering insight into how the function behaves for this input.

Algebraic Expressions

An algebraic expression is a combination of variables, constants, and operators like addition, subtraction, multiplication, and division. Polynomial expressions are a specific type of algebraic expressions that involve powers of variables with non-negative integer exponents.

Understanding algebraic expressions is crucial when simplifying polynomial substitutions and evaluations, as seen in the given exercise. Things to keep in mind:

• Variables represent unknowns and can be substituted with specific values or other expressions, like \( x \) with \( 2a^2 \) here.
• Exponents indicate how many times a variable or number is multiplied by itself.
• Terms are parts of the expression separated by plus or minus signs.”

When working with algebraic expressions, always ensure to simplify by combining like terms. Like terms have the same variable raised to the same power. Simplification not only makes expressions more manageable but also helps in accurately evaluating any algebraic form you are working with. In the exercise's final expression \( 8a^4 - 10a^2 + 4 \), each term represents a part of the polynomial's behavior, shaped by the replaced inputs.