solve the following in the real number system
x^3 + 8x^2 + 10x - 15 = 0
Please help me!
Since it is hard to factor a general cubic, look for easy roots. They will be among ±1,3,5,15
A little synthetic division shows that we have
(x+3)(x^2 + 5x - 5)
Now you can use the quadratic formula to get the other two roots:
(-5 ± 3√5)/2
by inspection, x=-3 is a solution.
-27+72-30-15=0
so, divide (x+3) into x^3+8x^2+10x-15 and get the quadratic. Then solve it with the quadratic equation.
I get x^2+5x-5 check that.
Thank you I think I understand it better with Bob's answer.
To solve the equation x^3 + 8x^2 + 10x - 15 = 0, you can use the Rational Root Theorem along with synthetic division or factoring to find the roots.
1. Apply the Rational Root Theorem: The Rational Root Theorem states that if the equation has a rational root, it can be expressed as p/q, where p is a factor of the constant term (-15) and q is a factor of the leading coefficient (1). Thus, the possible rational roots are: ±1, ±3, ±5, ±15.
2. Test the possible rational roots using synthetic division: Divide the polynomial by each possible rational root using synthetic division to check if it produces a remainder of zero. Start with p/q = 1.
(1) | 1 8 10 -15
|___ 1 9 19
----------------
1 9 19 4
Since the remainder is not zero, 1 is not a root. Repeat the process with the remaining possible roots until you find a root that produces a remainder of zero.
(-1) | 1 8 10 -15
|___ -1 -7 -3
----------------
1 7 3 -18
The remainder is not zero, so -1 is not a root.
(3) | 1 8 10 -15
|___ 3 33 99
----------------
1 11 43 84
Again, the remainder is not zero, so 3 is not a root.
(5) | 1 8 10 -15
|___ 5 65 375
----------------
1 13 75 360
Still, the remainder is not zero, so 5 is not a root.
(-3) | 1 8 10 -15
|___ -3 -15 15
----------------
1 5 -5 0
Finally, by dividing the polynomial by -3, we get a remainder of zero. Therefore, -3 is a root of the equation.
3. Write the equation in factored form: The equation (x^3 + 8x^2 + 10x - 15 = 0) can be rewritten as (x - (-3)) * (x^2 + 5x + 5) = 0.
4. Solve for the remaining roots: Set each factor equal to zero and solve for x.
x - (-3) = 0
x = -3
x^2 + 5x + 5 = 0
Using the quadratic formula, x = (-5 ± √(5^2 - 4*1*5))/2
x = (-5 ± √(25-20))/2
x = (-5 ± √5)/2
So, the real number solution to the equation x^3 + 8x^2 + 10x - 15 = 0 is x = -3, (-5 + √5)/2, and (-5 - √5)/2.