Finding the Other Polynomial Using HCF and LCM

This problem involves finding an unknown polynomial when its Highest Common Factor (HCF), Least Common Multiple (LCM), and one of the polynomials are given. The key property we will use is the relationship between the HCF and LCM of two polynomials and the product of the polynomials themselves.

Relationship Between HCF, LCM, and Polynomials

For any two polynomials, let's call them \(P_1(x)\) and \(P_2(x)\), and their HCF and LCM, the following relationship holds true:

\(\text{HCF}(P_1(x), P_2(x)) \times \text{LCM}(P_1(x), P_2(x)) = P_1(x) \times P_2(x)\)

We can rearrange this formula to find the unknown polynomial \(P_2(x)\):

\(P_2(x) = \frac{\text{HCF} \times \text{LCM}}{P_1(x)}\)

Applying the Formula to Find the Other Polynomial

Given:

• HCF = \((x + 3)\)
• LCM = \((x^3 - 9x^2 - x + 105)\)
• One polynomial \(P_1(x)\) = \((x^2 - 4x - 21)\)

We need to find the other polynomial \(P_2(x)\).

First, let's factor the given polynomial \(P_1(x) = x^2 - 4x - 21\). We look for two numbers that multiply to -21 and add up to -4. These numbers are -7 and +3.

So, \(P_1(x) = (x - 7)(x + 3)\).

Now, substitute the given values and the factored \(P_1(x)\) into the formula for \(P_2(x)\):

\(P_2(x) = \frac{(x + 3) \times (x^3 - 9x^2 - x + 105)}{(x - 7)(x + 3)}\)

We can cancel out the common factor \((x + 3)\) from the numerator and the denominator (assuming \(x \neq -3\)):

\(P_2(x) = \frac{x^3 - 9x^2 - x + 105}{x - 7}\)

To find \(P_2(x)\), we need to perform polynomial division, dividing \(x^3 - 9x^2 - x + 105\) by \(x - 7\).

Polynomial Division Steps

Let's perform the long division:

Divide \(x^3\) by \(x\): We get \(x^2\). Multiply \(x^2\) by \((x - 7)\): \(x^3 - 7x^2\). Subtract this from \(x^3 - 9x^2 - x + 105\):

\((x^3 - 9x^2) - (x^3 - 7x^2) = -2x^2\)

Bring down the next term, \(-x\). We now have \(-2x^2 - x\).

Divide \(-2x^2\) by \(x\): We get \(-2x\). Multiply \(-2x\) by \((x - 7)\): \(-2x^2 + 14x\). Subtract this from \(-2x^2 - x\):

\((-2x^2 - x) - (-2x^2 + 14x) = -x - 14x = -15x\)

Bring down the next term, \(+105\). We now have \(-15x + 105\).

Divide \(-15x\) by \(x\): We get \(-15\). Multiply \(-15\) by \((x - 7)\): \(-15x + 105\). Subtract this from \(-15x + 105\):

\((-15x + 105) - (-15x + 105) = 0\)

The remainder is 0. The quotient is \(x^2 - 2x - 15\).

Therefore, the other polynomial \(P_2(x)\) is \(x^2 - 2x - 15\).

Conclusion

By using the fundamental relationship between HCF, LCM, and the product of two polynomials, and performing polynomial division, we found the other polynomial.

The other polynomial is \(x^2 - 2x - 15\).

Revision Table: HCF and LCM of Polynomials

Additional Information: Polynomial Factorisation and Division

Understanding polynomial factorisation and polynomial long division is crucial for solving problems involving HCF and LCM of polynomials.

• Polynomial Factorisation: Breaking down a polynomial into a product of simpler polynomials (usually linear or irreducible quadratic factors). Methods include factoring out common terms, grouping, difference of squares, sum/difference of cubes, and using the quadratic formula.
• Polynomial Long Division: A method used to divide one polynomial by another polynomial of a lower or equal degree. It is similar to numerical long division and helps in finding factors or simplifying rational expressions. If a polynomial \(P(x)\) divided by \((x - a)\) gives a remainder of 0, then \((x - a)\) is a factor of \(P(x)\), and \(x = a\) is a root.

In this problem, we factored the known polynomial and used polynomial division to find the unknown polynomial.