Problem 27 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The value of \( p(x^2 + 3) \) is \( 3x^4 + 16x^2 + 26 \).

Step 1: Substitute \( x^2 + 3 \) into \( p(x) \)

The problem asks for \( p(x^2 + 3) \). Start by writing the given polynomial \( p(x) = 3x^2 - 2x + 5 \). To find \( p(x^2 + 3) \), substitute \( x^2 + 3 \) for \( x \) in the expression for \( p(x) \).

Step 2: Expand \( 3(x^2 + 3)^2 \)

First, compute \((x^2 + 3)^2\). This is equal to \((x^2 + 3)(x^2 + 3)\), which simplifies to \(x^4 + 6x^2 + 9\). Now multiply this result by 3: \[ 3(x^2 + 3)^2 = 3(x^4 + 6x^2 + 9) = 3x^4 + 18x^2 + 27 \]

Step 3: Expand \(-2(x^2 + 3)\)

Now compute \(-2(x^2 + 3)\). Distribute the -2 across the terms inside the parentheses:\[ -2(x^2 + 3) = -2x^2 - 6 \]

Step 4: Add the results together

Combine the results from Step 2 and Step 3, along with the constant term from \(p(x)\), which is 5. Combine like terms:\[ 3x^4 + 18x^2 + 27 - 2x^2 - 6 + 5 \].

Step 5: Simplify the expression

Finally, simplify the expression by combining like terms:\[ 3x^4 + (18x^2 - 2x^2) + (27 - 6 + 5) = 3x^4 + 16x^2 + 26 \].

Key Concepts

Polynomial Evaluation

When we talk about polynomial evaluation, we mean finding the value of a polynomial at a certain point or expression. Polynomials are expressions that comprise variables raised to whole number powers and multiplied by coefficients. Evaluating a polynomial involves replacing the variable in the polynomial with a given number or expression and then simplifying it to find the final value.

• In a polynomial like \( p(x) = 3x^2 - 2x + 5 \), "evaluating" involves calculating the polynomial's value for a specific input such as \( x = 2 \) or any other expression, like \( x^2 + 3 \).
• This process helps us determine what the polynomial's output is, given an input.

Understanding how to evaluate polynomials is crucial in math, as it lays the groundwork for solving more complex equations and problems.

Substitution Method

The substitution method involves replacing a variable with another expression to facilitate the evaluation of a polynomial or equation. This technique is particularly useful when dealing with nested expressions or when simplifying complex algebraic terms.

• For example, in our given problem, we substitute \( x \, \) in \( p(x) = 3x^2 - 2x + 5 \), with the expression \( x^2 + 3 \).
• It is achieved by replacing every instance of \( x \) in the polynomial with the given expression. In this case, every \( x \) becomes \( x^2 + 3 \).

By using this substitution method, we effectively change the problem into something manageable and followable, simplifying complex algebraic computations.

Polynomial Expansion

Polynomial expansion is the process of expanding expressions that have been condensed or simplified. When polynomials involve terms like \((x+a)^2\), they need to be expanded to perform arithmetic operations more easily.

• For the expression \((x^2 + 3)^2\), expansion involves multiplying \((x^2 + 3)(x^2 + 3)\).
• The resulting polynomial is \(x^4 + 6x^2 + 9\), which is the expanded form of \((x^2 + 3)^2\).

By expanding the polynomial, you simplify the process of combining like terms later. This expansion step allows easier manipulation and completion of further calculations to arrive at the final polynomial result.

Simplifying Expressions

Simplifying expressions is all about combining like terms to make the expression as concise as possible. It's the last stage in evaluating or manipulating polynomials, where you gather terms that can be merged to provide a simpler form.

• For instance, after evaluating and expanding the polynomial \( p(x^2 + 3) \), we have \( 3x^4 + 18x^2 + 27 - 2x^2 - 6 + 5 \).
• We then combine like terms: \( 18x^2 \) and \(-2x^2\) produce \( 16x^2 \), and \( 27 - 6 + 5 \) simplifies to 26.
• Thus, the expression simplifies to \( 3x^4 + 16x^2 + 26 \).

Simplification is key in ensuring expressions are easy to interpret, compare, and manipulate in various mathematical contexts. The process aids in decluttering and reaching the simplest form of the result possible.