Problem 68 - Algebra and Trigonometry

Question

\(65-70\) m Simplify the fractional expression. (Expressions like these arise in calculus.) $$ \frac{(x+h)^{3}-7(x+h)-\left(x^{3}-7 x\right)}{h} $$

Step-by-Step Solution

Answer

The simplified expression is \(3x^2 + 3xh + h^2 - 7.\)

Step 1: Expand the Cube

We start by expanding the cube of the binomial \[(x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3.\]

Step 2: Expand and Simplify

Next, we substitute the expanded form into the numerator. So the numerator becomes \[(x^3 + 3x^2h + 3xh^2 + h^3) - 7(x + h) - (x^3 - 7x).\]Expanding \(-7(x+h)\), this becomes \(-7x - 7h\).

Step 3: Combine Like Terms

Next, we combine like terms in the numerator. Starting with \[x^3 + 3x^2h + 3xh^2 + h^3 - 7x - 7h - (x^3 - 7x),\] the \(x^3\) terms will cancel out:\[3x^2h + 3xh^2 + h^3 - 7x - 7h + 7x.\]This simplifies to \[3x^2h + 3xh^2 + h^3 - 7h.\]

Step 4: Factor out 'h' from the Numerator

Notice each term in the numerator contains 'h'. Factor 'h' out:\[h(3x^2 + 3xh + h^2 - 7).\]

Step 5: Simplify the Fraction

Finally, cancel the 'h' from the numerator and the denominator, resulting in:\[3x^2 + 3xh + h^2 - 7.\]

Key Concepts

Expanding Binomials

Expanding binomials is essential when simplifying expressions like the one given in the exercise. Binomials are expressions with two terms, such as \((x + h)\). When we need to expand \((x + h)^3\), we apply the binomial theorem or simply use the distributive property repeatedly.

• Start by expanding the first power: \((x + h)^1 = x + h\).
• Then move to the square: \((x + h)^2 = (x+h)(x+h) = x^2 + 2xh + h^2\).
• Finally, multiply the square by \((x + h)\) once more: \((x + h)^3 = (x^2 + 2xh + h^2)(x+h)\).

Continuing this distribution process results in:
\[(x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\].
This expansion is crucial as it prepares the expression for further operations, like subtracting or combining similar terms.

Combining Like Terms

After expanding binomials, the exercise requires simplifying the expression by combining like terms. This step involves a careful look at the expression to gather terms with common factors or those sharing the same variables raised to the same powers.
In our exercise, after the initial expansion, you're left with several terms:
\[x^3 + 3x^2h + 3xh^2 + h^3 - 7x - 7h - (x^3 - 7x)\].

• First, identify and cancel identical terms. The \(x^3\) terms cancel each other out.
• Next, group the \(-7x\) and \(+ 7x\), which also cancel each other out.

After this cancellation, you are left with:
\[3x^2h + 3xh^2 + h^3 - 7h\].
Combining like terms makes the expression simpler and paves the way for factoring.

Factoring Polynomials

Factoring polynomials is a method of writing a polynomial expression as a product of simpler terms. In this exercise, each term of the remaining polynomial \(3x^2h + 3xh^2 + h^3 - 7h\) includes the variable \(h\).
To factor, you extract the common variable \(h\) from each term:
\[h(3x^2 + 3xh + h^2 - 7)\].

• This transformation simplifies the expression, preparing it for further simplification.
• Factoring is often paired with simplification, as seen here, where the \(h\) in the factor allows the division to reduce the expression further.

Ultimately, factoring transforms complex polynomials into manageable pieces, allowing for cancelation and reduction, as happens in this exercise where the entire expression becomes:
\[3x^2 + 3xh + h^2 - 7\].