Find the product
3[x-(- 1/3+2i�ã14/3)][x-(-1/3-2i�ã14/3)]
To find the product of the given expression, we need to simplify it step by step.
The given expression is:
3[x-(- 1/3+2i�ã14/3)][x-(-1/3-2i�ã14/3)]
Step 1: Simplify the innermost expressions inside the square brackets.
-1/3 + 2i�ã14/3 can be written as -1/3 + 2i√(14)/3
-1/3 - 2i�ã14/3 can be written as -1/3 - 2i√(14)/3
Therefore, the expression becomes:
3[x - (-1/3 + 2i√(14)/3)][x - (-1/3 - 2i√(14)/3)]
Step 2: Apply the distributive property by multiplying each term inside the square brackets by 3.
3 * x = 3x
3 * -(-1/3 + 2i√(14)/3) = 3/3 + 6i√(14)/3 = 1 + 2i√(14)
3 * x = 3x
3 * -(-1/3 - 2i√(14)/3) = 3/3 - 6i√(14)/3 = 1 - 2i√(14)
The expression now simplifies further to:
(3x + 1 + 2i√(14))(3x + 1 - 2i√(14))
Step 3: Apply the FOIL method to multiply the two binomials.
FOIL stands for First, Outer, Inner, Last.
(3x + 1 + 2i√(14))(3x + 1 - 2i√(14)) = (9x^2 + 3x + 6ix√(14) + 3x + 1 + 2i√(14) - 6ix√(14) - 2i^2(14))
Simplifying further:
9x^2 + 6x + 3x + 1 + 2i√(14) - 6ix√(14) - 2i^2(14)
Step 4: Simplify the expression by resolving the imaginary terms.
Recall that i^2 = -1.
Therefore, i^2(14) = -14.
The expression now becomes:
9x^2 + 6x + 3x + 1 + 2i√(14) - 6ix√(14) - 2(-14)
Simplifying further:
9x^2 + 9x + 1 + 2i√(14) - 6ix√(14) + 28
Finally, the product of the given expression is:
9x^2 + 9x + 29 + (2 - 6i)√(14)
So, the simplified form of the product is 9x^2 + 9x + 29 + (2 - 6i)√(14).