Problem 41 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The value is \(-x^6 + x^3 + 2x^2 + 4x + 2\).

Step 1: Evaluate r(x+1)

To find \(r(x+1)\), replace every instance of \(x\) in the function \(r(x)\) with \(x+1\). So, we have:\[r(x+1) = (x+1)^3 + (x+1) + 1\]Now expand \((x+1)^3\) using the binomial theorem:\[(x+1)^3 = x^3 + 3x^2 + 3x + 1\]Substituting back into the equation:\[r(x+1) = x^3 + 3x^2 + 3x + 1 + x + 1 + 1\]Combine like terms to simplify:\[r(x+1) = x^3 + 3x^2 + 4x + 3\]

Step 2: Evaluate r(x^2)

Next, substitute \(x^2\) into the function \(r(x)\) to find \(r(x^2)\), replacing \(x\) with \(x^2\):\[r(x^2) = (x^2)^3 + x^2 + 1\]Simplify the expression by calculating \((x^2)^3\):\[(x^2)^3 = x^6\]So the expression becomes:\[r(x^2) = x^6 + x^2 + 1\]

Step 3: Subtract r(x^2) from r(x+1)

With \(r(x+1)\) and \(r(x^2)\) known, subtract \(r(x^2)\) from \(r(x+1)\):\[r(x+1) - r(x^2) = (x^3 + 3x^2 + 4x + 3) - (x^6 + x^2 + 1)\]Distribute the negative sign across \(r(x^2)\):\[r(x+1) - r(x^2) = x^3 + 3x^2 + 4x + 3 - x^6 - x^2 - 1\]Combine the like terms:\[r(x+1) - r(x^2) = -x^6 + x^3 + 2x^2 + 4x + 2\]

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in algebra that involves finding the output of a function for a specific input. It's like asking, "What value does the function yield when we plug in this number?" In our exercise, we evaluate functions with different expressions substituted for \( x \). By manipulating inputs within the function, we can understand how variables transform outputs.

To evaluate \( r(x+1) \), we substitute \( x+1 \) for \( x \) in the function \( r(x) = x^3 + x + 1 \). This means replacing every instance of \( x \) with \( x+1 \), giving us \( r(x+1) = (x+1)^3 + (x+1) + 1 \). Similarly, for \( r(x^2) \), we substitute \( x^2 \) for \( x \), resulting in \( r(x^2) = (x^2)^3 + x^2 + 1 \).

Evaluating functions with such replacements helps us explore transformations within polynomial functions. This step is crucial for calculations and understanding various scenarios in algebra.

Binomial Theorem

The binomial theorem is vital when expanding expressions involving powers, especially in polynomial functions. It allows us to break down expressions like \( (x+1)^3 \) into simpler, manageable terms.

Here's how it works: The theorem states that \( (a+b)^n \) can be expanded into a sum involving terms of the form \( \binom{n}{k} a^{n-k} b^k \). For \( (x+1)^3 \), it results in:
\[ (x+1)^3 = \binom{3}{0} x^3 + \binom{3}{1} x^2 \cdot 1 + \binom{3}{2} x \cdot 1^2 + \binom{3}{3} 1^3 \]Computing these binomial coefficients (like \( \binom{3}{1} = 3 \)) results in \( x^3 + 3x^2 + 3x + 1 \).

Using the binomial theorem simplifies complex algebraic expressions and is especially handy for multiplying or expanding polynomials. It is a treasure trove of insights into polynomial expansions!

Algebraic Expressions

Algebraic expressions involve variables and constants combined using mathematical operations. Understanding how to manipulate and simplify them is a key skill in algebra.

In our step-by-step solution, we see algebraic expressions being simplified through substitution and combination of like terms. Let's break it down:

• Substitution: Replacement of variables with specific expressions, like \( x+1 \) or \( x^2 \) for \( x \).
• Expansion: Using the binomial theorem to expand terms like \( (x+1)^3 \) into individual components.
• Combination: Organizing similar terms such as \( 3x^2 + x^2 \) into a single term (\( 4x^2 \) here).

Mastering these procedures lets us take complex expressions and transform them into more digestible forms. This process is critical when solving or simplifying polynomial functions and expressions.