Problem 45 - Precalculus

Question

Show that the slope of the line passing through the two points \(\left(a, a^{3}\right)\) and \(\left(x, x^{3}\right)\) is \(x^{2}+a x+a^{2} .\) Hint: You'll need to use difference of cubes factoring from intermediate algebra. If you need a review, see Appendix B.4.

Step-by-Step Solution

Answer

The slope is \(x^2 + ax + a^2\).

Step 1: Understand the Slope Formula

The slope of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, our points are \((a, a^3)\) and \((x, x^3)\).

Step 2: Apply the Slope Formula

Substitute the coordinates of the points into the slope formula: \[ m = \frac{x^3 - a^3}{x - a}. \]

Step 3: Simplify Using Difference of Cubes

Recall the difference of cubes formula: \(x^3 - a^3 = (x - a)(x^2 + ax + a^2)\). Substitute this into the expression for the slope: \[ m = \frac{(x-a)(x^2 + ax + a^2)}{x - a}. \]

Step 4: Cancel Common Factors

The \((x-a)\) term can be cancelled from the numerator and the denominator because it's a common factor, assuming \(x eq a\). This simplifies the expression to: \[ m = x^2 + ax + a^2. \]

Step 5: Conclusion

Finally, the slope of the line through \((a, a^3)\) and \((x, x^3)\) is indeed \(x^2 + ax + a^2\), which verifies the given statement.

Key Concepts

Difference of Cubes

The concept of the difference of cubes is a useful algebraic identity. It allows us to simplify expressions involving the subtraction of two cubed terms. The formula for the difference of cubes is expressed as:

• \(x^3 - a^3 = (x - a)(x^2 + ax + a^2)\)

This formula breaks down the difference between the cubes of two numbers into a product of a linear factor \((x - a)\) and a quadratic trinomial \((x^2 + ax + a^2)\). This factorization is crucial, particularly in problems involving simplification or finding slopes, as seen in algebra and calculus courses.
To apply the difference of cubes, first identify if the expression fits the form \(x^3 - a^3\). If it does, you can directly use the identity to factor the expression, making it simpler for further algebraic manipulation.

Slope Formula

The slope formula is a fundamental concept in algebra and geometry. It describes the steepness of a line connecting two points in a coordinate plane. Mathematically, the slope \(m\) is defined as follows:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

For any two points \( (x_1, y_1) \) and \( (x_2, y_2) \), this formula calculates the ratio of the vertical change \((y_2 - y_1)\) to the horizontal change \((x_2 - x_1)\).
The slope indicates both the direction and the steepness of the line:

• A positive slope means the line ascends from left to right.
• A negative slope means the line descends from left to right.
• A zero slope indicates a horizontal line.
• An undefined slope results from a vertical line where \(x_1 = x_2\).

Understanding and applying the slope formula is essential for interpreting and graphing linear equations, helping visualize relationships between two variables.

Algebraic Simplification

Algebraic simplification involves reducing complex expressions to their simplest form. This process makes it easier to work with and understand mathematical relationships. In calculus and algebra, simplification often involves factoring, canceling terms, and combining like terms.
For example, in our exercise, we simplified an expression using the difference of cubes formula. Starting with:

• \(\frac{x^3 - a^3}{x - a} = \frac{(x-a)(x^2 + ax + a^2)}{x - a}\)

We noticed that \((x-a)\) appeared in both the numerator and the denominator. Assuming \(x eq a\), we canceled this common factor to simplify the expression to \(x^2 + ax + a^2\).
This simplification strategy makes solving complex equations more manageable and is applicable in various mathematical contexts, ensuring that calculations are both accurate and efficient.