Problem 46 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the polynomial whose roots are 1,2 and 3 .

Step-by-Step Solution

Answer

The polynomial whose roots are 1, 2, and 3 is \(P(x) = x^3 - 6x^2 + 11x - 6\).

Step 1: Write the polynomial with its factors based on the roots

Since the polynomial has roots 1, 2, and 3, we can write the polynomial as a product of factors using the property mentioned above: P(x) = A(x-1)(x-2)(x-3)

Step 2: Expand the polynomial

Now, expand the polynomial by multiplying the factors: P(x) = A(x-1)(x-2)(x-3) First, multiply (x-1) and (x-2): (x-1)(x-2) = x^2 - 3x + 2 Now, multiply the result by (x-3): P(x) = A(x^2 - 3x + 2)(x-3) P(x) = A(x^3 - 3x^2 + 2x - 3x^2 + 9x - 6x + 6)

Step 3: Combine like terms and simplify

Combine like terms and simplify the polynomial: P(x) = A(x^3 - 6x^2 + 11x - 6)

Step 4: Choose the value of A

Since we are not given any constraints on the polynomial, we can choose any value for A. However, the simplest choice is to set A=1, as this will not affect the roots of the polynomial and will produce the simplest polynomial form. So, the polynomial is: P(x) = x^3 - 6x^2 + 11x - 6

Key Concepts

Roots of a Polynomial

Understanding the concept of polynomial roots is fundamental in algebra. The roots of a polynomial are values of the variable that make the polynomial equal to zero. To find these roots, we factor the polynomial into linear factors. For example, if we know the roots of a polynomial are 1, 2, and 3, we can express the polynomial using these roots. This means that the polynomial can be written as a product of factors:

• \( (x-1) \) because at \( x = 1 \), this factor becomes zero.
• \( (x-2) \) because at \( x = 2 \), this factor becomes zero.
• \( (x-3) \) because at \( x = 3 \), this factor becomes zero.

So, the polynomial with roots 1, 2, and 3 is written as \( P(x) = (x-1)(x-2)(x-3) \). Roots help us understand the behavior and graph of polynomial functions as they are the points where the graph intersects the x-axis. When dealing with polynomials, identifying the roots first can often simplify the process of factoring and expanding.

Expanding Polynomials

Expanding a polynomial involves removing parentheses to express the polynomial as a sum of terms. This is done by multiplying the factors together to simplify the expression. In our example, we have the polynomial \( (x-1)(x-2)(x-3) \). To expand, we start by multiplying the first two factors:

• Multiply \( (x-1) \) and \( (x-2) \) to get \( x^2 - 3x + 2 \).

Then, take that result and multiply by the third factor \( (x-3) \). Doing so requires us to distribute each term:

• \( (x^2 - 3x + 2) \times (x-3) \)
• This eventually simplifies to: \( x^3 - 6x^2 + 11x - 6 \)

This expanded form is more practical for computing values and analyzing the polynomial's degree and leading coefficient. It makes understanding the polynomial's end behavior and curvature easier. Knowing how to expand correctly is essential for simplifying polynomials and solving more complex equations.

Polynomial Simplification

Simplifying polynomials is a skill useful for making polynomials easier to work with. Once a polynomial is expanded, we need to simplify it by combining like terms. Like terms are terms that have the same variable raised to the same power. For instance, in the expanded polynomial \( x^3 - 6x^2 + 11x - 6 \), there are no like terms to combine further, but often there will be.Simplifying consists of:

• Seeing if any terms can be combined.
• Checking if they can be factored again for further simplification.
• Ensuring that the polynomial is expressed in its cleanest form.

In some problems, you might have a coefficient like \( A \) in \( A(x^3 - 6x^2 + 11x - 6) \). If \( A \) is not determined by the problem, a common practice is to set \( A = 1 \) to make the expression straightforward.Keeping the polynomial in a simplified form is beneficial for ease of interpretation, and it helps when solving for roots or analyzing the behavior of the graph.