Problem 10 - Calculus for JEE ADVANCED

Question

We have $$ f(x)+g(x)+h(x)=\frac{x^{2}-4 x+17-4 x-2}{x^{2}+x-12} $$

Step-by-Step Solution

Answer

\(f(x) + g(x) + h(x) = \frac{x - 5}{x + 4}\).

Step 1: Simplify the Numerator

Consider the numerator of the given expression: \(x^2 - 4x + 17 - 4x - 2\). Simplify this by combining like terms:- Combine \(-4x - 4x\) to get \(-8x\).- Combine \(17 - 2\) to get \(15\). Therefore, the simplified numerator is \(x^2 - 8x + 15\).

Step 2: Identify the Denominator

The denominator in the expression is given by \(x^2 + x - 12\). There is nothing to simplify here as it is presented. The exercise requires the expression \(\frac{x^2 - 8x + 15}{x^2 + x - 12}\).

Step 3: Factor the Numerator

Factorize the simplified numerator \(x^2 - 8x + 15\):- Find two numbers that multiply to 15 and add up to -8. These numbers are -3 and -5. Hence, it factors to \((x - 3)(x - 5)\).

Step 4: Factor the Denominator

Factorize the denominator \(x^2 + x - 12\):- Find two numbers that multiply to -12 and add to 1. These are 4 and -3.- Hence, the denominator factors to \((x + 4)(x - 3)\).

Step 5: Cancel Common Factors

After factoring both the numerator and the denominator, the expression becomes:\[\frac{(x - 3)(x - 5)}{(x + 4)(x - 3)}.\]Cancel the common factor \((x - 3)\) from both the numerator and the denominator, resulting in:\[\frac{x - 5}{x + 4}.\]

Step 6: State the Final Function

The given expression \(f(x) + g(x) + h(x)\) simplifies to the function:\[\frac{x - 5}{x + 4}.\] Therefore, \(f(x) + g(x) + h(x) = \frac{x - 5}{x + 4}\).

Key Concepts

Factoring Quadratics

Factoring quadratics is the process of breaking down a quadratic expression into simpler factors that, when multiplied together, yield the original expression. It often involves finding two numbers that multiply to give the constant term and add up to the coefficient of the linear term.

For example, consider the quadratic expression

• \(x^2 - 8x + 15\).
• To factor this, we need two numbers that multiply to 15 and add up to -8.
• After identifying these numbers as -3 and -5, we can write the expression as \((x - 3)(x - 5)\).

This method helps in simplifying further operations, such as simplifying rational expressions later.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing them to their simplest form. This often means factoring the numerator and the denominator as much as possible. Once factored, terms that appear in both the numerator and the denominator can be simplified.

Take the given rational expression:

• \(\frac{x^2 - 8x + 15}{x^2 + x - 12}\).

First, factor the numerator and the denominator as much as possible. This helps make the next step, canceling common factors, ready. Identifying the factors:

• Numerator: \((x - 3)(x - 5)\)
• Denominator: \((x + 4)(x - 3)\)

With these, we are prepared to see and cancel out common factors.

Canceling Common Factors

Canceling common factors is a key technique used to simplify rational expressions. After factoring both the numerator and the denominator, look for terms that are common to both and can thus be canceled out, simplifying the expression.

For instance, consider:

• \(\frac{(x - 3)(x - 5)}{(x + 4)(x - 3)}\).
• The factor \((x - 3)\) appears in both the numerator and the denominator.
• Since these are the same, they can be canceled, simplifying the expression to: \(\frac{x - 5}{x + 4}\).

Canceling these factors helps to reduce the expression to its simplest form, making it easier to work with in equations or functions.