Respuesta :
We are basically asked to perform the product of expressions that contain complex numbers a s roots of polynomials. So it is essential to identify the complex root in order to facilitate our products that otherwise could be really time consuming.
First product:
(x + 3 + 5i) (x + 3 - 5i) we are going to write as: ((x +3) +5i) ((x+3) - 5i). So this is going to be interpreted as a difference of squares (a + b) ( a - b) = a^2 - b^2:
((x +3) +5i) ((x+3) - 5i) = (x+3)^2 - (5i)^2 = (x+3)^2 - (-25) = (x+3)^2 + 25
We are using the fact that (i)^2 = -1 to convert - 25 (i)^2 = + 25
Now we solve the square of the binomial (x+3) , that is:
(x+ 3)^2 = x^2 + 6x + 9
Now this combined with + 25 gives: x^2 + 6x + 34 This is one of the options in blue background shown at the tops (the fourth one starting from the left)
Second product:
(x - 4i) (x + 4i) which is an easirer product than the one above because we can solve it just using difference of squares: (a - b) (a + b) = a^2 - b^2:
(x - 4i) (x + 4i) = x^2 - (4i)^2 = x^2 - 16 (i)^2 = x^2 +16
This is the second expression in blue background from left to right.
Third product:
(x - 6 + i) (x - 6 - i) for which we apply the same technique as in the first product we solve above:
(x - 6 + i) (x - 6 - i) = ((x - 6) + i) ((x - 6) - i) = (x-6)^2 - (i)^2 = (x-6)^2 - (-1) = (x-6)^2 + 1 = x^2 - 12 x + 36 + 1 = x^2 - 12 x + 37 which is the very first expression in blue background in the image you uploaded.
Olease move the bue background boxes so as to match the indicated products.
