Problem 55 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{(x+h)^{2}-3(x+h)-\left(x^{2}-3 x\right)}{h}$$

Step-by-Step Solution

Answer

The simplified expression is \(2x + h - 3\).

Step 1: Expand the Squared Term

First, expand the squared expression \((x+h)^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\). For \((x+h)^2\), substitute \(a = x\) and \(b = h\), yielding: \[(x+h)^2 = x^2 + 2xh + h^2.\]

Step 2: Distribute the Multiplied Term

Next, distribute the multiplication over the subtraction \(-3(x+h)\): \[-3(x+h) = -3x - 3h.\]

Step 3: Substitute and Simplify the Numerator

Substitute the expanded and distributed expressions back into the original expression:\[(x^2 + 2xh + h^2) - 3x - 3h - (x^2 - 3x).\] Simplify by combining like terms in the numerator:- Cancel out \(x^2\)- Collect like terms:\[2xh + h^2 - 3h + 3x = 2xh + h^2 - 3h - 0 + 0.\]

Step 4: Simplify the Expression

The expression within the numerator becomes:\[2xh + h^2 - 3h.\] Relabel the entire expression as:\[\frac{2xh + h^2 - 3h}{h}.\] Factor out \(h\) from the numerator:\[\frac{h(2x + h - 3)}{h}.\]

Step 5: Cancel Out the Common Term

With \(h\) factored out, divide each term by \(h\), resulting in the simplified expression:\[2x + h - 3.\] Ensure \(h eq 0\) to avoid division by zero.

Key Concepts

Expanding Expressions

When tasked with expanding expressions in algebra, we often look to transform them into a more workable form. Consider the expression \( (x+h)^2 \). Here, the goal is to "expand" or open up the expression by applying a formula. Specifically, we utilize the formula for squaring a binomial: \( (a+b)^2 = a^2 + 2ab + b^2 \). By substituting \( a = x \) and \( b = h \), the expression expands to \( x^2 + 2xh + h^2 \).

• Step 1: Recognize the squared term needing expansion.
• Step 2: Apply the binomial expansion formula.
• Step 3: Rewrite the expression in its expanded form.

Recognizing and applying these steps will simplify complex problems, making them much easier to handle.

Factoring

Factoring is the process of breaking down an expression into simpler multipliers that can be easily managed or canceled. In our example:\[(2xh + h^2 - 3h)\]we factor by identifying the greatest common factor, which is \( h \).
To factor the expression:

• Step 1: Identify the common factor. Here, each term includes an \( h \), so we factor out \( h \).
• Step 2: Write the expression as \( h(2x + h - 3) \).

This process not only simplifies the expression but also reveals solutions more clearly by reducing it to its simplest form.

Like Terms

Combining like terms is an essential skill in simplifying algebraic expressions. This occurs when you add or subtract terms that share the same variable raised to the same power.
In the expression\(x^2 + 2xh + h^2 - 3x - 3h - (x^2 - 3x)\):

• Combine: Terms \( x^2 \) terms cancel each other out as they have opposite coefficients.
• Identify: The term \(3x\) from both the positive and negative instances cancels.
• Simplify: What's left is \( 2xh + h^2 - 3h\).

Understanding how to handle like terms will streamline the simplification process, helping you see the core of the problem quicker.

Polynomial Simplification

Simplifying polynomials involves a series of steps including expanding, combining like terms, and factoring as we've done here. It requires attention to detail and a structured approach. Let's look at the expression we've been working on:
\[\frac{2xh + h^2 - 3h}{h}\]

• Step 1: Make sure to use only necessary operations, like factoring out h.
• Step 2: Cancel the common \( h \) in the numerator and denominator.
• Conclusion: Arrive at the simplest form, \( 2x + h - 3 \), ensuring \( h \) is not zero.

Each step builds on the previous, solidifying your understanding of polynomials and enhancing your problem-solving toolkit.