Problem 62 - Applied Calculus

Question

For each function, find and simplify \(f(x+h)\). $$ f(x)=3 x^{2}-5 x+2 $$

Step-by-Step Solution

Answer

The simplified expression for \(f(x+h)\) is \(3x^2 - 5x + 6xh + 3h^2 - 5h + 2\).

Step 1: Substitute \(x+h\) in the function

Start by replacing every occurrence of \(x\) in the function \(f(x) = 3x^2 - 5x + 2\) with \(x+h\). This gives us: \[ f(x+h) = 3(x+h)^2 - 5(x+h) + 2 \]

Step 2: Expand the squared term

Expand \((x+h)^2\). This can be done using the formula \((a+b)^2 = a^2 + 2ab + b^2\). Therefore: \[ (x+h)^2 = x^2 + 2xh + h^2 \] Now substitute this back into the expression for \(f(x+h)\): \[ f(x+h) = 3(x^2 + 2xh + h^2) - 5(x+h) + 2 \]

Step 3: Distribute the constants into the expanded terms

Distribute the constant \(3\) across the terms inside the parentheses: \[ 3(x^2 + 2xh + h^2) = 3x^2 + 6xh + 3h^2 \] Now the function becomes: \[ f(x+h) = 3x^2 + 6xh + 3h^2 - 5(x+h) + 2 \]

Step 4: Expand and simplify \(-5(x+h)\)

Expand \(-5(x+h)\) by distributing \(-5\):\[ -5(x+h) = -5x - 5h \] Substitute back into \(f(x+h)\):\[ f(x+h) = 3x^2 + 6xh + 3h^2 - 5x - 5h + 2 \]

Step 5: Combine like terms

Combine any like terms to simplify the expression:The terms \(3x^2\) and \(-5x\) can be combined:\[ 3x^2 - 5x \]And that yields the simplified expression for \(f(x+h)\):\[ f(x+h) = 3x^2 - 5x + 6xh + 3h^2 - 5h + 2 \]

Step 6: Conclusion: Simplified expression

The simplified expression for \(f(x+h)\) is:\[ f(x+h) = 3x^2 - 5x + 6xh + 3h^2 - 5h + 2 \]

Key Concepts

Polynomials

Polynomials are a fundamental part of algebra that consist of variables raised to different powers, usually combined with coefficients. They can range from simple terms like a constant or a single variable term, to more complex expressions like quadratic or cubic polynomials. When we talk about a polynomial function, such as the one in our exercise, it’s often presented in the format of a sum of terms, like:

• Constant Term: "+2" in our example
• Linear Term: "-5x" in our example
• Quadratic Term: "3x^2" in our example

To analyze or manipulate these expressions, understanding their structure is crucial. Each part of the polynomial contributes differently to its overall behavior. Quadratic polynomials like the one given in the exercise, play a vital role in various areas such as real-world applications in economics, physics, and even biology.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and operations (like addition and multiplication) represented in a meaningful way. For instance, our function, which is given by the expression \(3x^2 - 5x + 2\), is a typical algebraic expression.
These expressions represent a wide range of mathematical concepts and provide a compact way to convey complex operations. They can be simple with just a few terms or highly complex with many different types of operations.

• The expression can be transformed by substituting terms, as we replace 'x' with 'x+h'.
• Algebraic expressions allow for systematic manipulation, such as expanding or factoring.
• They form the basis for evaluating and understanding function transformations.

The key is learning how these expressions can be systematically managed, leading to various mathematical solutions across numerous fields.

Expansion and Simplification

Expansion and simplification are critical processes in handling algebraic expressions, particularly polynomials. Expansion involves rewriting expressions in an extended form, while simplification is about reducing them to their simplest form.
For our exercise:

• We expand using algebraic identities like \((a+b)^2 = a^2 + 2ab + b^2\), which transform terms clearly.
• Once expanded, the expression contains more parts than it initially had. Each term needs careful attention.
• Simplification incorporates combining like terms — these are terms with the same variables raised to the same powers.

This transformation not only aids in solving problems but also enhances our understanding of the relationships between terms. Through strategic expansion and simplification, complex polynomials are turned into manageable forms that are easier to work with or analyze. This skill is particularly useful in higher mathematics and practical real-world scenarios.