Solve the polynomial equation by factoring
18x^3 -15x^2 - 9x =0
I got as far as
3x(6x^2 - 5x - 3) = 0
Thanks
if it seems hard to factor, check the discriminant: b^2-4ac
In this case, that is 5^2 - 4(6)(-3) = 97
That is not a perfect square, so you cannot factor it with rational coefficients.
Run it through the quadratic formula to confirm.
To solve the polynomial equation 18x^3 - 15x^2 - 9x = 0 by factoring, you correctly factored out 3x from the equation. Now, let's factor the quadratic expression (6x^2 - 5x - 3) further.
To factor the quadratic expression, we need to find two numbers whose product is equal to the product of the coefficient of x^2 (6) and the constant term (-3) and whose sum is equal to the coefficient of x (-5). In this case, we can use the numbers -3 and 1.
Therefore, the factored form of the quadratic expression is:
6x^2 - 5x - 3 = (2x - 3)(3x + 1)
Now, substitute this back into the factored equation:
3x(6x^2 - 5x - 3) = 0
3x(2x - 3)(3x + 1) = 0
Now, we can solve for x by setting each factor equal to zero:
3x = 0 (from the factor 3x = 0)
2x - 3 = 0 (from the factor 2x - 3 = 0)
3x + 1 = 0 (from the factor 3x + 1 = 0)
Solving these equations, we find:
x = 0 (from 3x = 0)
x = 3/2 (from 2x -3 = 0)
x = -1/3 (from 3x + 1 =0)
Therefore, the solutions to the original polynomial equation 18x^3 - 15x^2 - 9x = 0 are x = 0, x = 3/2, and x = -1/3.
To solve the polynomial equation 18x^3 - 15x^2 -9x = 0 by factoring, you have correctly factored out the common factor of 3x:
3x(6x^2 - 5x - 3) = 0
To continue factoring, we need to find two binomials that multiply to give us the trinomial 6x^2 - 5x - 3. Let's tackle this step by step:
1. First, we need to look for pairs of numbers that multiply to give the constant term (-3) and add up to give the coefficient of the linear term (-5). In this case, the pairs that satisfy these conditions are (-3, 1) and (1, -3).
2. Next, we need to split the middle term (-5x) using the pairs we found. We rewrite the trinomial as follows:
6x^2 - 3x + x - 3 = 0
3. We can now factor the trinomial by grouping:
(6x^2 - 3x) + (x - 3) = 0
3x(2x - 1) + (x - 3) = 0
4. Finally, we have two expressions that have a common binomial factor of (2x - 1). We can factor it out:
3x(2x - 1) + (x - 3) = 0
(2x - 1)(3x + 1) = 0
So, the factored form of the polynomial equation 18x^3 - 15x^2 - 9x = 0 is:
3x(2x - 1)(3x + 1) = 0
Now we have three factors: 3x = 0, 2x - 1 = 0, and 3x + 1 = 0. Solve each equation separately to find the values of x.