Answer:
{1, (-1±√17)/2}
Step-by-step explanation:
There are formulas for the real and/or complex roots of a cubic, but they are so complicated that they are rarely used. Instead, various other strategies are employed. My favorite is the simplest--let a graphing calculator show you the zeros.
___
Descartes observed that the sign changes in the coefficients can tell you the number of real roots. This expression has two sign changes (+-+), so has 0 or 2 positive real roots. If the odd-degree terms have their signs changed, there is only one sign change (-++), so one negative real root.
It can also be informative to add the coefficients in both cases--as is, and with the odd-degree term signs changed. Here, the sum is zero in the first case, so we know immediately that x=1 is a zero of the expression. That is sufficient to help us reduce the problem to finding the zeros of the remaining quadratic factor.
__
Using synthetic division (or polynomial long division) to factor out x-1 (after removing the common factor of 4), we find the remaining quadratic factor to be x²+x-4.
The zeros of this quadratic factor can be found using the quadratic formula:
a=1, b=1, c=-4
x = (-b±√(b²-4ac))/(2a) = (-1±√1+16)/2
x = (-1 ±√17)2
The zeros are 1 and (-1±√17)/2.
_____
The graph shows the zeros of the expression. It also shows the quadratic after dividing out the factor (x-1). The vertex of that quadratic can be used to find the remaining solutions exactly: -0.5 ± √4.25.
__
The given expression factors as ...
4(x -1)(x² +x -4)
Answer:
1/2 base x height
Step-by-step explanation:
Recall the double/half angle formulas:
![\cos^2\dfrac x2=\dfrac{1+\cos x}2](https://tex.z-dn.net/?f=%5Ccos%5E2%5Cdfrac%20x2%3D%5Cdfrac%7B1%2B%5Ccos%20x%7D2)
![\sin^2\dfrac x2=\dfrac{1-\cos x}2](https://tex.z-dn.net/?f=%5Csin%5E2%5Cdfrac%20x2%3D%5Cdfrac%7B1-%5Ccos%20x%7D2)
We're given
, and since
is between π/2 and π, we expect
to be negative. So from the Pythagorean identity, we find
![\sin^2u+\cos^2u=1\implies\cos u=-\sqrt{1-\sin^2u}=-\dfrac7{25}](https://tex.z-dn.net/?f=%5Csin%5E2u%2B%5Ccos%5E2u%3D1%5Cimplies%5Ccos%20u%3D-%5Csqrt%7B1-%5Csin%5E2u%7D%3D-%5Cdfrac7%7B25%7D)
Also, we know
will fall between π/4 and π/2, so both
and
will be positive. Then we find
![\cos\dfrac u2=\sqrt{\dfrac{1+\cos u}2}=\dfrac35](https://tex.z-dn.net/?f=%5Ccos%5Cdfrac%20u2%3D%5Csqrt%7B%5Cdfrac%7B1%2B%5Ccos%20u%7D2%7D%3D%5Cdfrac35)
![\sin\dfrac u2=\sqrt{\dfrac{1-\cos u}2}=\dfrac45](https://tex.z-dn.net/?f=%5Csin%5Cdfrac%20u2%3D%5Csqrt%7B%5Cdfrac%7B1-%5Ccos%20u%7D2%7D%3D%5Cdfrac45)
and it follows that
![\tan\dfrac u2=\dfrac{\sin\frac u2}{\cos\frac u2}=\dfrac43](https://tex.z-dn.net/?f=%5Ctan%5Cdfrac%20u2%3D%5Cdfrac%7B%5Csin%5Cfrac%20u2%7D%7B%5Ccos%5Cfrac%20u2%7D%3D%5Cdfrac43)
The correct answer is D. k= 3
Explain
First , point ( x,y) by factor k
It will become (x/k,y)
The vertex is (-3,-3) of f(x)
And the other vertex is (-1,-3) of g(x)
-1,-3)= (-3/3 ,3)
So the dilation factor is 3
(-1,-3) = (-3 x 1/3 ,3)
Factor is k
g(x)= f(1/3 x)
So k = 3
Because the graph have shrunk f(x) by factor 3
I Hope this make sense to you
Glad I can help you :D