How can you check if 3+i is a zero of the equation x^4-6x^3+6x^2+24x-40?
One way is to substitute x=3+i into the equation and evaluate:
x^4-6x^3+6x^2+24x-40.....(1)
=(3+i)^4-6(3+i)^3+6(3+i)²+24(3+i)-40
=28+96i - 6(18+26i) + 6(8+6i) + 24(3+i) -40
=96i-156i+36i+24i + 28-108+48+72-40
=0i+0
=0
Therefore (3+i) is a zero of the given equation.
Note that the complex conjugate of the given zero is also a zero if the original equation has real coefficients. So as a bonus, 3-i is also a zero of (1).
Another way is to factorize the given expression on the left-hand side, namely
(x-2)*(x+2)*(x^2-6*x+10)
Use the quadratic formula to find the two complex roots in the last term of the factorization to give 3±i as the remaining zeroes.
To check if 3+i is a zero of the equation x^4 - 6x^3 + 6x^2 + 24x - 40, we can substitute it into the equation and see if it equals zero.
Step 1: Substitute 3+i for x in the equation.
Substituting 3+i into the equation, we get:
(3+i)^4 - 6(3+i)^3 + 6(3+i)^2 + 24(3+i) - 40
Step 2: Expand and simplify the expression.
Using the binomial expansion formula for (a+b)^4, we can simplify the expression:
(81 + 108i - 18 - 12i + 1) - 6(27 + 9i - 9 - 3i + 1) + 6(9 + 6i + 1) + 72 + 24i - 40
Simplifying further, we get:
(-10 + 96i) - 6(28 + 6i) + 6(16 + 6i) + 72 + 24i - 40
Step 3: Combine like terms.
Combining the terms, we have:
-10 + 96i - 168 - 36i + 96 + 36i + 72 + 24i - 40
Step 4: Simplify the expression.
Simplifying further, we get:
66 + 120i
Step 5: Check if the result is zero.
Since 66 + 120i is not zero, we can conclude that 3+i is not a zero of the equation x^4 - 6x^3 + 6x^2 + 24x - 40.
Therefore, after evaluating the expression, we can see that 3+i is not a zero of the equation.