+234 1) find the sec θ for and angle whose teminal side contains that poing (-3,4)
2) give x-5 is a factor of x^3 + 9x^2 - 37x - 165 what are the other factors
3) solve 16^x = (1/4)^(x+4)
4) simplify (9x + 27)/x^2 - 6x + 8) ÷ (6x + 18)/(4x^2 - 16)
5) simplify (2/x) + 3/(x-1) ÷ 1/(2x-2)
6) what is g(f(x)) if f(x) = 2x^2 - x and g(x) = x^(-1/2)
+128460 Gonna eat supper, Blank.....hang on and I'll answer these in just a few minutes...
+128460 1) find the sec θ for and angle whose teminal side contains the point (-3,4)
Sec = r /x
r = sqrt [ (-3)^2 + 4^2 ] = sqrt (25) = 5
So....sec θ = 5 / - 3
The angle will be in the 2nd quadrant
To find this....use the secant inverse....
arcsec (5/-3) ≈ 126.9° = θ
+128460 2) give x-5 is a factor of x^3 + 9x^2 - 37x - 165 what are the other factors
We can use sythetic division to dwtermine the remaining polynomial
5 [ 1 9 - 37 - 165 ]
5 70 165
_____________________
1 14 33 0
The remaining polynomial is x^2 + 14 + 33
Factoring this we have that
(x + 11) ( x + 3) = the remaining factors !!!
+234 So basically we take the answer of the provided factor and use that as the multiplier in the syntetic divison to get a quadratic function that we use any way to factor down into the final factors? So how would this work if there was a quadratic to the 5th power? The same thing?
+128460 We would do the same thing.....one thing though....the remaining polynomial may or may not be factorable
Fortunately....this one WAS capable of being factored....!!!!
[ We might have to use the Rational Zeroes Theorem to find the other possible roots if the remaining polynomial doesn't factor easily ]
+128460 3) solve 16^x = (1/4)^(x+4)
Note that 16 = 4^2
And 1/4 = 4^(-1)
So we have
(4^2)^x = (4^(-1) )( x + 4) using a law of exponents
4^(2x) = (4)^ (-x - 4) we have the same bases....so....we can solve for the exponents
2x = -x - 4 add x to both sides
3x = - 4 divide both sides by 3
x = -4/3
+128460 4) simplify (9x + 27)/x^2 - 6x + 8) ÷ (6x + 18)/(4x^2 - 16)
Factor tops/bottoms
9 (x + 3) 6 (x + 3)
__________ ÷ ____________ flip the second fraction and multiply
(x -4)( x - 2) (x + 4)(x - 4)
9(x + 3) (x + 4) ( x - 4)
__________ * _____________
(x - 4) ( x - 2) 6(x + 3)
9(x + 3) * (x + 4) ( x - 4)
___________________
6 (x + 3) * (x -4) ( x - 2)
3 (x + 4) 3x + 4
_______ = _____
2 (x - 2) 2x - 4
+128460 5) simplify (2/x) + 3/(x-1) ÷ 1/(2x-2)
Easiest to take this in "pieces"
2 3 2 (x - 1) 2(x - 1)
_ + ____ = ____ * ______ = _______
x (x - 1) x 3 3x
So now...we have
2(x - 1) 1 2(x - 1) (2x- 2) 2(x - 1) * 2 (x - 1) 4(x - 1)^2
______ + ______ = _________ * ______ = _________________ = _________
3x (2x - 2) 3x 1 3x 3x
