The HCF and the LCM of two polynomials are 3x + 1 and 30x 3 + 7x 2 - 10x - 3 respectively. If one polynomial is 6x 2 + 5x + 1, then what is the other polynomial?
15x 2- 4x - 3
Finding the Other Polynomial using HCF and LCM
This problem involves using the fundamental relationship between the Highest Common Factor (HCF), the Least Common Multiple (LCM), and two polynomials. Let the two polynomials be \(P_1(x)\) and \(P_2(x)\).
The key property connecting these is:
HCF \(\times\) LCM = \(P_1(x) \times P_2(x)\)
In this question, we are given:
- HCF \( = 3x + 1\)
- LCM \( = 30x^3 + 7x^2 - 10x - 3\)
- One polynomial \(P_1(x) = 6x^2 + 5x + 1\)
We need to find the other polynomial, \(P_2(x)\).
From the property, we can express \(P_2(x)\) as:
\(P_2(x) = \frac{\text{HCF} \times \text{LCM}}{P_1(x)}\)
Substituting the given expressions:
\(P_2(x) = \frac{(3x + 1)(30x^3 + 7x^2 - 10x - 3)}{6x^2 + 5x + 1}\)
Factoring the Given Polynomial
First, let's factorize the given polynomial \(P_1(x) = 6x^2 + 5x + 1\).
We look for two numbers that multiply to \(6 \times 1 = 6\) and add up to \(5\). These numbers are \(3\) and \(2\).
\(6x^2 + 5x + 1 = 6x^2 + 3x + 2x + 1\)
Group the terms:
\(= (6x^2 + 3x) + (2x + 1)\)
Factor out common factors from each group:
\(= 3x(2x + 1) + 1(2x + 1)\)
Factor out the common binomial factor \((2x + 1)\):
\(= (3x + 1)(2x + 1)\)
So, \(P_1(x) = (3x + 1)(2x + 1)\).
Simplifying the Expression for the Other Polynomial
Now substitute the factored form of \(P_1(x)\) back into the equation for \(P_2(x)\):
\(P_2(x) = \frac{(3x + 1)(30x^3 + 7x^2 - 10x - 3)}{(3x + 1)(2x + 1)}\)
Since \((3x+1)\) is a common factor in the numerator and the denominator (and it is the HCF, which must divide both polynomials), we can cancel it out:
\(P_2(x) = \frac{30x^3 + 7x^2 - 10x - 3}{2x + 1}\)
To find \(P_2(x)\), we need to perform polynomial division: divide \(30x^3 + 7x^2 - 10x - 3\) by \(2x + 1\).
Polynomial Division
Let's perform the long division:
| \(15x^2\) | \(-\) \(4x\) | \(-\) \(3\) | ||
| \(2x + 1\) | \(30x^3\) | \(+ 7x^2\) | \(-\) \(10x\) | \(-\) \(3\) |
| \(-\) \((30x^3\) | \(+ 15x^2)\) | |||
| \(0\) | \(-\) \(8x^2\) | \(-\) \(10x\) | ||
| \(-\) \((-\) \(8x^2\) | \(-\) \(4x)\) | |||
| \(0\) | \(-\) \(6x\) | \(-\) \(3\) | ||
| \(-\) \((-\) \(6x\) | \(-\) \(3)\) | |||
| \(0\) | \(0\) |
Here are the steps of the polynomial division:
- Divide \(30x^3\) by \(2x\) to get \(15x^2\). Write \(15x^2\) in the quotient.
- Multiply \(15x^2\) by \((2x + 1)\) to get \(30x^3 + 15x^2\).
- Subtract \((30x^3 + 15x^2)\) from \((30x^3 + 7x^2)\). This gives \(-8x^2\). Bring down the next term, \(-10x\).
- Divide \(-8x^2\) by \(2x\) to get \(-4x\). Write \(-4x\) in the quotient.
- Multiply \(-4x\) by \((2x + 1)\) to get \(-8x^2 - 4x\).
- Subtract \((-\) \(8x^2 - 4x)\) from \((-\) \(8x^2 - 10x)\). This gives \(-6x\). Bring down the last term, \(-3\).
- Divide \(-6x\) by \(2x\) to get \(-3\). Write \(-3\) in the quotient.
- Multiply \(-3\) by \((2x + 1)\) to get \(-6x - 3\).
- Subtract \((-\) \(6x - 3)\) from \((-\) \(6x - 3)\). This gives \(0\).
The quotient is \(15x^2 - 4x - 3\).
Therefore, the other polynomial \(P_2(x)\) is \(15x^2 - 4x - 3\).
Conclusion
The other polynomial is \(15x^2 - 4x - 3\).
Revision Table: HCF, LCM, and Polynomials
| Concept | Description | Key Property |
| HCF (Highest Common Factor) | The polynomial of highest degree that divides two or more polynomials exactly. | HCF \(\times\) LCM = Product of the polynomials |
| LCM (Least Common Multiple) | The polynomial of lowest degree that is a multiple of two or more polynomials. | |
| Relationship between HCF and LCM | For any two polynomials \(P_1\) and \(P_2\), the product of their HCF and LCM is equal to the product of the polynomials themselves. | \(P_1 \times P_2 = \text{HCF}(P_1, P_2) \times \text{LCM}(P_1, P_2)\) |
Additional Information: Polynomial Division
Polynomial long division is an algorithm used to divide a polynomial by another polynomial of the same or lower degree. It is similar to the process of long division with numbers.
- It helps in factoring polynomials when one factor is known.
- It is used to simplify rational expressions (fractions with polynomials).
- If a polynomial \(P(x)\) is divided by \(D(x)\) and the remainder is \(R(x)\) and the quotient is \(Q(x)\), then \(P(x) = D(x) \times Q(x) + R(x)\).
- In the case of the HCF-LCM problem, the division should result in a remainder of zero because LCM is a multiple of the polynomial, and the product of HCF and LCM is a multiple of both polynomials.
Similar Questions
If (x 2- 1) is a factor of ax 4+ bx 3+ cx 2+ dx + e, then which one of the following is correct?
If α, β and γ are the zeros of the polynomial f(x) = ax 3+ bx 2+ cx + d, then α 2+ β 2+ γ 2is equal to
Let f(x) and g(x) be two polynomials (with real coefficients) having degree 3 and 4 respectively. What is the degree of f(x) g(x)
If α and β are the two zeros of the polynomial 25x 2– 15x + 2, then what is a quadratic polynomial whose zeros are (2α) -1 and (2β) -1 ?
The expression \(\frac{{\left( {{x^3} - 1} \right)\left( {{x^2} - 9x + 14} \right)}}{{\left( {{x^2} + x + 1} \right)\left( {{x^2} - 8x + 7} \right)}}\) simplifies to
If (6x + 4y) / (6x - 4y) = 8/6 then what is the value of x 2 / y 2?
What is the sum of the linear factors (in x and y) of the expression
2x 2 + xy - 3y 2 ?
What should be added to \(\frac{1}{{(x - 2)(x - 4)}}\) to get \(\frac{2x - 5}{{(x^2-5x+6)(x - 4)}}\) ?
What is (x - a) (x - b) (x - c) equal to?
For what value of k can the expression x 3+ kx 2– 7x + 6 be resolved into three linear factors?
Important Questions from Polynomials
If y 2= y + 7, then what is the value of y 3?
Factorize x 2- y 2- 9z 2+ 6yz
If one of the zeros of the polynomial x 3+ ax 2+ bx + c is - 1, then the product of other two zeros is equal to :
If a(a + b + c) 2 = 1792; b(a + b + c) 2 = 1536; c(a + b + c) 2 = 768, then what will be the value of b?
If x = 3 so, what is the value of x 2 + 2x + 5 ?
