HCF and LCM of two polynomials are (x + 3) and (x 3- 9x 2- x + 105). If one of the two polynomials is (x 2- 4x - 21), then the other is
x 2- 2x - 15
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
| Concept | Description | Key Property |
|---|---|---|
| HCF (Highest Common Factor) | The polynomial of the highest degree that divides two or more polynomials exactly. Found by taking common factors with the lowest power. | HCF \(\times\) LCM = Product of the polynomials |
| LCM (Least Common Multiple) | The polynomial of the lowest degree that is a multiple of two or more polynomials. Found by taking all factors with the highest power. |
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.
Similar Questions
If (x + k) is the HCF of x 2+ 5x + 6 and x 2+ 8x + 15, then what is the value of k?
A floor of a big hall has dimensions 30 m 60 cm and 23 m 40 cm. It is to be paved with square tiles of same size. What is the minimum number of tiles required ?
The LCM of two prime numbers p and q is 2231, where p > q. What is the value of p - q ?
Three runners are running in a circular track, and they complete one round in 20, 30 and 35 minutes respectively. When will they next meet at the starting point ?
What is the least perfect square which is divisible by 3, 4, 5, 6 and 7?
If (x - k) is the HCF of x 2+ ax + b and x 2+ cx + d, then what is the value of k?
HCF of two numbers is 12. Which one of the following can never be their LCM?
X, Y and Z start at some point and same time in the same direction to run around a circular stadium. X completes a round in 252 seconds, Y in 308 seconds and Z in 198 seconds. After what time will they meet again at the starting point?
What is the LCM of the polynomials x 3+ 3x 2+ 3x + 1, x 3+ 5x 2+ 5x + 4 and x 2+ 5x + 4?
The product of two integers p and q, where p > 60 and q > 60, is 7168 and their HCF is 16. The sum of these two integers is:
Important Questions from LCM and HCF
The HCF and LCM of two numbers are 12 and 72, respectively. If the ratio of the two numbers is 2 ∶ 3, then the larger of the two numbers is:
Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.
Joseph visits the club on every 5 th day, Harsh visits on every 24 th day, while Sumit visits on every 9 th day. If all three of them met at the club on a Sunday, then on which day will all three of them meet again?
What is the least number which when divided by 12,20 and 24 leaves in each case a remainder of 8?
Which of the following is a pair of co-primes?
