Problem 41 - Contemporary Precalculus

Question

Expand and simplify the rule of the function \(g .\) $$f(x)=x^{4}+3 x+5 \text { and } g(x)=f(x-2)$$

Step-by-Step Solution

Answer

Question: Expand and simplify the given function \(g(x) = f(x-2)\), where \(f(x) = x^4 + 3x + 5\). Answer: \(g(x) = x^4 - 4x^3 + 6x^2 - x + 15\)

Step 1: Substitute \(x-2\) into the function \(f(x)\).

Begin by replacing all instances of \(x\) in the function \(f(x)\) with \((x-2)\). This will give us the function \(g(x)\): $$g(x) = f(x-2) = (x-2)^4 + 3(x-2) + 5$$

Step 2: Expand \((x-2)^4\)

We can expand \((x-2)^4\) using the binomial theorem: $$ (x-2)^4 = x^4 - 4x^3 + 6x^2 - 4x + 16 $$

Step 3: Expand and combine terms in \(g(x)\)

Now, expand the remaining terms in \(g(x)\) and combine like terms: \begin{align*} g(x) &= x^4 - 4x^3 + 6x^2 - 4x + 16 + 3(x-2) + 5 \\ &= x^4 - 4x^3 + 6x^2 - 4x + 16 + 3x - 6 + 5 \\ &= x^4 - 4x^3 + 6x^2 - x + 15 \end{align*}

Step 4: Write the final simplified form of \(g(x)\)

The final simplified form of \(g(x)\) is: $$g(x) = x^4 - 4x^3 + 6x^2 - x + 15$$

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool used in algebra to expand expressions that are raised to a power. It allows us to break down expressions like \((x-2)^4\) into a series of terms. Each term follows a specific pattern involving binomial coefficients, which can be found using combinations.

• First, identify the two parts of the binomial. Here, they are \(x\) and \(-2\).
• The expression is expanded by applying the formula for the binomial theorem: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
• In our example, \(a = x\), \(b = -2\), and \(n = 4\). Therefore, we plug these values into the formula:

Applying this pattern gives us \(x^4 - 4x^3 + 6x^2 - 4x + 16\). Notice how each power of \(x\) decreases, while the power of \(-2\) increases, following the binomial theorem's structure. This systematic approach makes expanding binomials both predictable and manageable.

Function Composition

Function composition involves taking one function and applying it to the result of another function. When we see \(g(x) = f(x-2)\), it's an example of composing functions. Here's how it works with our specific example:

• Start with the function \(f(x) = x^4 + 3x + 5\).
• To compose \(g(x)\), substitute \((x-2)\) for every occurrence of \(x\) in \(f(x)\). This gives a new function, \(g(x)\).
• For our equation, it is \(g(x) = (x-2)^4 + 3(x-2) + 5\).

By composing functions, we essentially modify the input of the first function \(f\) with the expression \(x-2\). This substitution transforms the context and results in a new function \(g\), which is tailor-made to respect the composition.

Polynomial Simplification

Simplifying polynomials involves expanding and combining all similar terms to write the expression in its most concise form. Looking at the polynomial expression for \(g(x)\), we simplify it by performing a few steps:

• First, distribute and expand each term fully.
• Then, rearrange them into like terms: constants, \(x\) terms, \(x^2\) terms, etc.
• Combine coefficients of the same terms to streamline the expression.

For example, in our polynomial \((x-2)^4 + 3(x-2) + 5\), we first expanded \((x-2)^4\) and then distributed \(3(x-2)\). After that, we collected like terms, resulting in \(x^4 - 4x^3 + 6x^2 - x + 15\).By following these steps, we eliminate redundancy and arrive at the simplest possible expression for the polynomial. This makes it easier to understand and work with in future mathematical operations.