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solve the cubic equation
2x^3-11x^2+18x-8=0
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#thanks
They probably expect you to use factoring, so try factors of 8
so x = ±1, ± 2 , ± 4 would be values to start with,
Sure enough on the third try, x = 2 works
Thus x-2 is a factor and by synthetic division,
2x^3-11x^2+18x-8=0
(x-2)(2x^2 -7x + 4) = 0
so one root is x = 2,
find the other two by using the quadratic formula on the second factor.
To solve the cubic equation 2x^3 - 11x^2 + 18x - 8 = 0, we can use a method called "factorization" or "rational root theorem".
Step 1: Factorize the constant term. The constant term is -8, and we need to find its factors. The possible factors of -8 are ±1, ±2, ±4, and ±8.
Step 2: Use the rational root theorem to find the potential rational roots. The rational root theorem states that any rational root of the equation must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In our case, the leading coefficient is 2.
The possible rational roots are:
±1/1, ±1/2, ±2/1, ±2/2, ±4/1, ±4/2, ±8/1, ±8/2.
Step 3: Test the potential rational roots using synthetic division or substitution to find the roots.
By substituting the potential rational roots into the equation, we find that x = 1 is a root of the equation. So, (x - 1) is a factor of the equation.
Step 4: Use polynomial long division to divide 2x^3 - 11x^2 + 18x - 8 by (x - 1).
2x^2 - 9x - 8
----------------------
x - 1 | 2x^3 - 11x^2 + 18x - 8
- ( 2x^3 - 2x^2 )
----------------------
- 9x^2 + 18x
+ ( -9x^2 + 9x )
----------------------
0
The quotient is 2x^2 - 9x - 8.
Step 5: Factorize the quadratic equation 2x^2 - 9x - 8 using the factorization method, quadratic formula, or completing the square. Since the quadratic equation cannot be easily factorized, we will use the quadratic formula.
The quadratic formula is given as:
x = (-b ± √(b^2 - 4ac)) / (2a)
Using the formula, where a = 2, b = -9, and c = -8, we have:
x = (-(-9) ± √((-9)^2 - 4(2)(-8))) / (2(2))
x = (9 ± √(81 + 64)) / 4
x = (9 ± √(145)) / 4
The roots of the equation are:
x = (9 + √145) / 4
x = (9 - √145) / 4
Therefore, the solutions to the cubic equation 2x^3 - 11x^2 + 18x - 8 = 0 are x = 1, x = (9 + √145) / 4, and x = (9 - √145) / 4.
To solve the cubic equation 2x^3 - 11x^2 + 18x - 8 = 0, you can apply different methods such as synthetic division, factoring, or using the cubic formula.
Let's use the most common method, which is factoring by grouping:
Step 1: Group the terms in pairs to factor by grouping:
(2x^3 - 8) + (-11x^2 + 18x)
Step 2: Factor out the greatest common factor from each group:
2(x^3 - 4) - 1(11x^2 - 18x)
2(x - 2)(x^2 + 2x + 1) - x(11x - 18)
Step 3: Simplify:
2(x - 2)(x + 1)^2 - x(11x - 18)
Now, we have factored the cubic equation as 2(x - 2)(x + 1)^2 - x(11x - 18) = 0.
To find the solutions, set each factor equal to zero and solve for x:
First factor: x - 2 = 0
x = 2
Second factor: (x + 1)^2 = 0
x + 1 = 0
x = -1 (note that this is a repeated root)
Third factor: 11x - 18 = 0
11x = 18
x = 18/11
Therefore, the solutions to the cubic equation 2x^3 - 11x^2 + 18x - 8 = 0 are x = 2, x = -1 (repeated root), and x = 18/11.