I need someone to explain the steps in finding all zeroes of an equation
x^4-6x^2-7x-6=0
Steve I do not understand how to use the quadratic formula to find the other roots would you please explain?
This does not help me out.
To find all the zeroes of an equation, like x^4 - 6x^2 - 7x - 6 = 0, you can use a combination of factoring, synthetic division, and the quadratic formula. Here's a step-by-step breakdown of how you can find all the zeroes:
Step 1: Check for Rational Roots
Start by checking for any possible rational roots using the Rational Root Theorem. 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 (-6 in this case) and q is a factor of the leading coefficient (1 in this case).
In this example, the possible rational roots are ±1, ±2, ±3, ±6. To check if any of these are actual roots, substitute them into the equation to see if you get zero. For example, let's substitute x = 1:
(1)^4 - 6(1)^2 - 7(1) - 6 = 1 - 6 - 7 - 6 = -18 ≠ 0
Since x = 1 is not a root, continue checking the other possible rational roots. If you find a root, move to the next step. If not, move to Step 2.
Step 2: Factor The Equation
If you didn't find any rational roots in Step 1, you can try to factor the equation. Start by grouping terms and factor out common terms:
x^4 - 6x^2 - 7x - 6 = (x^4 - 6x^2) - (7x + 6)
= x^2(x^2 - 6) - (7x + 6)
= x^2(x^2 - 6) - (7x + 6)
The factor x^2 - 6 can be factored further as the difference of squares:
x^2 - 6 = (x^2 - 2^2)
= (x + 2)(x - 2)
Now substituting this back into the main equation:
(x + 2)(x - 2) - (7x + 6) = 0
Step 3: Solve for the Roots
You now have a factored equation. Set each factor equal to zero and solve for x:
x + 2 = 0 => x = -2
x - 2 = 0 => x = 2
So two of the zeroes of the equation are x = -2 and x = 2.
Step 4: Synthetic Division
Using synthetic division, divide the factored equation (x + 2)(x - 2) into the original equation to get a reduced quadratic equation.
Performing synthetic division:
2 | 1 0 -7 -6
-------------------
2 4 -6
-------------------
1 2 -3 -12
The reduced quadratic equation is: x^2 + 2x - 3 = 0
Step 5: Solve the Quadratic Equation
To find the remaining zeroes, solve the reduced quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 1, b = 2, and c = -3. Plugging these values into the quadratic formula:
x = (-2 ± √(2^2 - 4(1)(-3))) / (2(1))
= (-2 ± √(4 + 12)) / 2
= (-2 ± √16) / 2
= (-2 ± 4) / 2
So the remaining roots are:
x = (-2 + 4) / 2 => x = 1
x = (-2 - 4) / 2 => x = -3
Thus, the equation x^4 - 6x^2 - 7x - 6 = 0 has four zeroes: -2, 2, 1, and -3.
you know the rational roots, if any, are among ±1 ±2 ±3 ±6
so, try some and fond that -2 and 3 are roots.
dividing those out you find you have
(x+2)(x-3)(x^2 + x + 1)
find the other two roots using the quadratic formula