algebra precalculus – How to compare which interest rate is better compounded annual vs compounded 3 times a year

Having a little trouble with getting the answer to this question.

: How much better is the return on a 4% yearly interest rate investment that is compounded 3 times per year as opposed to compounded yearly?

I tried to set up the equation as :

10000(1.04)^n = 10000(1+.04/3)^3n n=1

then to compare the them :

10400/ 1.040535704 = 0.999485

I’m guessing I am not setting things up right…

I am suppose to get answer 1% – 1.5% better.

algebra precalculus – Prove that the square root of distinct prime numbers is irrational

Prove that if $p_1,…,p_k$ are distinct prime numbers, then $sqrt{p_1p_2…p_k}$ is irrational.

I do not usually prove theorems, so any hint is appreciated. I have taken a look at this and tried to repeat that argument over and over, but I messed up. Perhaps there is an easier to do it. Thanks in advance.

algebra precalculus – Unable to Solve a quiz question asked in mathematics exam ( Quantitative Aptitude)

I am self studying for an exam and I am unable to solve this quiz question.

Adding it’s image ->

enter image description here

I tried by finding numbers in the sentences but couldn’t find and I think that’s a wrong approach.

Can anyone please tell how to solve this question.

Answer is B.

algebra precalculus – How should I write the factors in polynomial division if I know the polynomials roots?

If a polynomial P(x) has a root at x=0.5 and I do polynomial division on P(x), would I divide it by (x-0.5) or (2x-1)? Are both equally valid? Furthermore, would it be okay to write the polynomial in factored form as P(x) = (x-0.5) * Q(x), or should I have integer coefficients?

algebra precalculus – Actuarial theory of interest question with effective discount

I have spent several hours trying to solve this problem. $A$ and $B$ both open up new bank accounts at time $0$. The principle for $A$ (the amount deposited at $t=0$) is $100$. The principle of $B$ is $50$ (the amount deposited at $t=0$). Each account earns an annual discount rate of $d$. The amount of interest earned in $A$ during the 11th period is equal to $X$. The amount of interest earned in $B$ during the 17th period is equal to $X$. Calculate X.

Given that we are dealing with effective discount rate, for $A$ and $B$ we have $a(t)^{-1} = (1-d)^t$ clearly the amount of interest earned is $(1-d)^t$ for both $A$ and $B$. This means that $(1-d)^{11}=(1-d)^{17}$
Is this the correct set up? if not why? Given that the principles are $100, 50$ we have:

$frac{100}{(1-d)^{11}}=frac{50}{(1-d)^{17}}$ that is if we are setting the amount functions equal to each other during the $17$ and $11$ period. I thought this would translate to $50(1-d)^{11} = 100(1-d)^{17}$ then substituting $X$ for $(1-d)^{17}$ and $(1-d)^{11}$ we have something that makes no sense $50X=100X$ which gives $X=0$ after solving. If I don’t substitute for $X$ I have $50(1-d)^{11}=100(1-d)^{17} to 50=100(1-d)^6$ which translates to $frac{1}{2}=(1-d)^{6}$ which gives a decimal less than one. The answer is $38.88$ I have no idea where I am going wrong. What am I missing? Am I even any where close on my thought processes?

Algebra precalculus – polynomial transformations and Vietas formulas

To let $ f (x) $ be a monic, cubic polynomial with $ f (0) = – $ 2 and $ f (1) = – $ 5. Is the sum of all solutions too $ f (x + 1) = 0 $ and to $ f big ( frac1x big) = 0 $ are equal to what is $ f (2) $?

From $ f (0) $ I got that $ f (x) = x ^ 3 + ax ^ 2 + bx-2 $ and from $ f (-1) = – $ 5 The $ a + b = -4 $ However, I'm not sure how to use the transform information to find it $ f (2). $ It appears that $ (x + 1) $ is a root for $ f (x + 1) $ and the same logic applies to $ f big ( frac1x big) $?

Should I use Vieta's here or what is the right way?

Algebra precalculus – Prove that $ X ^ n + aX ^ {n-1} + … + aX-1 $ in $ mathbb {Z} cannot be reduced.[X]$.

Prove that $ X ^ n + aX ^ {n-1} + … + aX-1 $ is irreducible in $ mathbb {Z} (X) $, Where $ n ge 2 $ and $ a in mathbb {Z} ^ {*} $.
To let $ f = X ^ n + aX ^ {n-1} + … + aX-1 $. I was able to show this through direct calculations $ f = (X-1 + a) (X ^ {n-1} + X ^ {n-2} + … + X + 1) -a $ and then I tried to assume that it is not irreducible, i.e. $ exists g, h in mathbb {Z} (X) $ so that $ f = g cdot h $ and $ deg g, deg h <n $.
That was not useful because I could only get that $ g (0) = 1 $ and $ h (0) = – 1 $ or $ g (0) = – 1 $ and $ h (0) = 1 $ (which could be obtained from $ f $Initial form). I also tried to take advantage of the fact that $ f (1-a) = – a $, but without success.

Algebra precalculus – Find the equation of the parabola that passes through point (2, -1), whose vertex is at (-7, 5), and that opens to the right.

I've seen several questions like this, but I want to know how to write the equation $ (y-k) ^ 2 = 4p (x-h) $ form instead of vertex form.

The question is this: Find the equation of the parabola that passes through point (2, -1), whose apex is at (-7, 5), and that opens to the right.

  1. When I insert the corner point, the equation is $ (y-5) ^ 2 = 4p (x + 7) $,
  2. I understand that if the parabola opens to the right, $ p> 0 $,

How would i find $ p $ but for this equation?

Algebra precalculus – solving after the closed form of the repetition relation, given by $ x_ {n + 1} = (x_ {n-1}) ^ 2 + n (x_ {n}) $

Problem: Get closed form of $ x_n $ using the following: $ x_ {n + 1} = (x_ {n-1}) ^ 2 + n (x_ {n}) $ ; $ x_0 = 3, x_1 = 4 $

Try: I understand $ x_n = n + 3 $ by observation after inserting values ​​of $ x_0 $ and $ x_1 $ to get $ x_2 = 5 $ and then $ x_3 = 6 $,

To prove this, I have proven this by induction $ x_n = n + 3 $ is a valid solution.

Provided $ x_ {k-1} = k + 2 $ and $ x_ {k} = k + 3 $ for an integer $ k $,
Clear $ x_0 $ and $ x_14 $ to meet the conditions.
Now, $ x_ {k + 1} = (k + 2) ^ 2-k (k + 3) = k + 4 $ as required; This completes the induction.

So we get $ x_n = n + 3 $ for all $ n $ in whole numbers.

Doubt: How do I show that this is the only solution?

Is there also an alternative method to derive this result? If so, please share, thanks.

Algebra precalculus – inequality in relation to non-negative numbers

I would like to demonstrate the following inequality:

$ text {max} {s_n ^ 2, s_ {n-1} ^ 2 } + 4 sum_ {i = 1} ^ ns_ {i-1} (s_i-s_ {i-1}) leq 4s_n ^ 2 $

where I follow from $ n + 1 $ real numbers, ALL of which are non-negative, and that $ s_0 leq s_1 leq … leq s_ {n-1} $ ($ s_n $ can be any non-negative real number).

I've tried to reorder this several times, but I'm stuck. Please help me if you can.