Find all real Solution:
2x^3-7x^2-98x-343
To find all real solutions of the equation 2x^3 - 7x^2 - 98x - 343 = 0, we can use the rational root theorem and synthetic division.
The rational root theorem states that if a rational number p/q is a root of a polynomial equation with integer coefficients, then p must be a factor of the constant term (in this case, -343) and q must be a factor of the leading coefficient (in this case, 2).
Step 1: Find the factors of the constant term (-343): ±1, ±7, ±49, ±343
Step 2: Find the factors of the leading coefficient (2): ±1, ±2
By trying out possible pairs of factors (p/q) using synthetic division, we can determine if any of them are roots of the equation.
Trying p/q = 1:
Perform synthetic division with (x - 1) as the divisor:
1 | 2 -7 -98 -343
| - 1 -6 -92
-----------------
2 -8 -104 -435
The remainder is not zero, so 1 is not a root.
Trying p/q = -1:
Perform synthetic division with (x + 1) as the divisor:
-1 | 2 -7 -98 -343
| -2 9 -2 96
-----------------
2 -8 -100 -247
The remainder is not zero, so -1 is not a root.
Trying p/q = 7:
Perform synthetic division with (x - 7) as the divisor:
7 | 2 -7 -98 -343
| 14 49 343
-----------------
2 7 -49 0
The remainder is zero, so 7 is a root.
Therefore, one real solution is x = 7.
To find the other two roots, we need to solve the quadratic equation obtained by dividing the original polynomial by (x - 7).
Dividing 2x^3 - 7x^2 - 98x - 343 by (x - 7), we get:
2x^2 + 21x + 49
Solving this quadratic equation, we can use factoring, completing the square, or the quadratic formula.
The quadratic factors as:
(2x + 7)(x + 7) = 0
Setting each factor to zero, we have:
2x + 7 = 0 --> x = -7/2
x + 7 = 0 --> x = -7
Therefore, the real solutions to the equation 2x^3 - 7x^2 - 98x - 343 = 0 are:
x = 7, -7/2, -7