7.
y = ( x + 8)^3 + 9 shifts the parent function - x^3 - to the left 8 units and up 3 units
The parent function has only one real zero and this translated function also has one real zero
So....the answer is 1
8. If we can add the coefficients and constant and get 0,, then 1 is a root
So....in the first one -1 -6 + 6 - 6 + 5 = -2 .....so it isn't this one
Look at the second one 1 + 6 - 6 + 6 - 5 = 2 not this one either
Look at the last one x^4 - 6x^3 + 6x^2 - 6x + 5
We can write x^4 - 6x^3 + 5x^2 + x^2 -6x + 5
Factor as x^2(x^2 - 6x + 5) + 1 (x^2 - 6x + 5)
(x^2 + 1) ( x^2 - 6x + 5)
(x^2 + 1) (x - 5) ( x - 1)
Notice that if we set the last two factors to 0 and solve for x....we get that x = 5 and x = 1
And these are the two zeroes.......so...the last one is correct
9.
If -1 + 4i and -1 - 4i are roots....we have that
[ x - (-1 + 4i) ] [ x - (-1 - 4i) ] = 0 expand this, NSS....watch the math!!!
[ x^2 - x (- 1 + 4i) - x(-1 - 4i) + (-1 + 4i) (-1 - 4i) ] = 0
[ x^2 + x - 4xi + x + 4xi + ( 1 + 4i - 4i - 16i^2 ] = 0
Remember that { -16i^2 = -16 * -1 = 16 }
[ x^2 + 2x + 1 + 16 ] = 0
[ x^2 + 2x + 17 ] = 0
