PLEASE SHOW WORK FOR ALL PROBLEMS, I WANT TO UNDERSTAND HOW TO DO IT, NOT JUST THE FINAL ANSWER :)
1. find all the zeros of the equation -3x^4+27x^2+1200=0
2. Use the Binomial Theorem to expand the binomial
(d-4b)^3
3. What is the equation of y=x^3 with the given transformations?
vertical stretch by a factor of 3, horizontal shift 4 units to the right, vertical shift 3 units down
THANK YOU SO MUCH :)
1. let t=x^2 (t>0)
the equation: -3t^2+27t+1200=0 ( you can solve this function by finding Delta)
<=>t=25 and t=-16( t>0)
<=>t=25 <=>x^2=25<=>x=-5,5
2. (a-b)^3=(a-b)(a-b)^2
=(a-b)(a^2-2ab+b^2)=a^3-3a^2b+3ab^2-b^3 ( then just apply)
3.stretch: 3x^3
right: 3(x-4)^3
down: 3(x-4)^3 - 3
1. To find the zeros of the equation -3x^4 + 27x^2 + 1200 = 0, we need to factor the equation and set each factor equal to zero, then solve for x.
First, let's try to factor out an x^2 from each term: x^2(-3x^2 + 27 + 1200/x^2) = 0.
Next, we can focus on the quadratic term -3x^2 + 27. Let's try factoring out a common factor of -3: -3(x^2 - 9). Now we have x^2(-3)(x^2 - 9 + 400/x^2) = 0.
The quadratic term x^2 - 9 can be factored using the difference of squares: (x - 3)(x + 3).
Now we have x^2(-3)(x - 3)(x + 3 + 400/x^2) = 0.
Setting each factor equal to zero:
x^2 = 0
x - 3 = 0
x + 3 = 0
x + 3 + 400/x^2 = 0
Solving these equations, we obtain the zeros of the given equation:
x = 0 (double root)
x = 3
x = -3
x^2 + 3x + 400 = 0
2. To expand the binomial (d - 4b)^3 using the Binomial Theorem, we can use the formula:
(a + b)^n = C(n, 0)a^n b^0 + C(n, 1)a^(n-1) b^1 + C(n, 2)a^(n-2) b^2 + ... + C(n, n-1)a^1 b^(n-1) + C(n, n)a^0 b^n
In our case, a = d, b = -4b, and n = 3.
Using the values in the formula, we have:
(d - 4b)^3 = C(3, 0)d^3 (-4b)^0 + C(3, 1)d^2 (-4b)^1 + C(3, 2)d^1 (-4b)^2 + C(3, 3)d^0 (-4b)^3
Simplifying further, we get:
(d - 4b)^3 = d^3 - 12d^2 b + 48db^2 - 64b^3
3. To find the equation of y = x^3 with the given transformations, we need to apply each transformation step by step.
a) Vertical stretch by a factor of 3: Multiply the function by the factor 3.
y = 3 * x^3
b) Horizontal shift 4 units to the right: Replace x with (x - 4).
y = 3 * (x - 4)^3
c) Vertical shift 3 units down: Subtract 3 from the function.
y = 3 * (x - 4)^3 - 3
This is the equation of the transformed function.