The number of roots common between the two equations x3 + 7x2 + 3x + 1...
To get the common roots, we need to find the intersecting points of the two equations.
So, subtract the second equation from the first, we get,
x^{2} - 3x + 2 = 0
The solution of this equation gives x = 1 and x = 2
Put this two values in the equations. Since, both of the values are not satisfying the two equations. It has no common roots.
Answer: 0
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The number of roots common between the two equations x3 + 7x2 + 3x + 1...
Introduction:
We are given two cubic equations and we need to find the number of common roots between them. The equations are:
1) x^3 + 7x^2 + 3x + 10 = 0
2) x^3 + 6x^2 + 6x + 8 = 0
Explanation:
Step 1: Factorizing the equations:
We can start by trying to factorize the equations. However, it is not easy to factorize cubic equations directly. So, we need to use other methods to find the common roots.
Step 2: Using the Rational Root Theorem:
The Rational Root Theorem states that if a polynomial equation has a rational root p/q, where p and q are integers with no common factors, then p must be a factor of the constant term and q must be a factor of the leading coefficient.
For equation 1: x^3 + 7x^2 + 3x + 10 = 0
- The constant term is 10 and its factors are ±1, ±2, ±5, ±10.
- The leading coefficient is 1 and its factors are ±1.
For equation 2: x^3 + 6x^2 + 6x + 8 = 0
- The constant term is 8 and its factors are ±1, ±2, ±4, ±8.
- The leading coefficient is 1 and its factors are ±1.
Step 3: Checking for common roots:
Now, we can check if any of the factors of the constant terms are common roots for both equations.
For equation 1:
- Checking ±1: Substitute x = ±1 in the equation, we get -1 + 7 + 3 - 10 = 0, which is not equal to 0.
- Checking ±2: Substitute x = ±2 in the equation, we get 8 + 28 + 6 + 10 = 52, which is not equal to 0.
- Checking ±5: Substitute x = ±5 in the equation, we get 125 + 175 + 15 + 10 = 325, which is not equal to 0.
- Checking ±10: Substitute x = ±10 in the equation, we get 1000 + 700 + 30 + 10 = 1740, which is not equal to 0.
For equation 2:
- Checking ±1: Substitute x = ±1 in the equation, we get 1 + 6 + 6 + 8 = 21, which is not equal to 0.
- Checking ±2: Substitute x = ±2 in the equation, we get 8 + 24 + 12 + 8 = 52, which is not equal to 0.
- Checking ±4: Substitute x = ±4 in the equation, we get 64 + 96 + 24 + 8 = 192, which is not equal to 0.
- Checking ±8: Substitute x = ±8 in the equation, we get 512 + 384 + 48 + 8 = 952, which is not equal to 0.
Conclusion:
After checking all the possible common roots, we can see that none of the
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