0
votes
0answers
17 views

Can I evaluate polynomials with prime numbers to find possible irreductible factors?

Let $p(x,y)$, $c(x,y)$ and $d(x,y)$ be two variable polynomials with integer coefficients which satisfy $p(x,y)=c(x,y)\cdot d(x,y)$. Given $m, n$ positive prime numbers and given $e(x,y)$ another ...
1
vote
3answers
31 views

Factor Cyclic Polynomial

Factor $(a+b)(b+c)(c+a)+abc$. I know this is a cyclic polynomial, but I don't know how to solve problems like this. What should I do?
3
votes
3answers
89 views

Factor $3x^2-11xy+6y^2-xz-4yz-2z^2$

This problem is from my Math Challenge II Algebra class, and it's really confusing. How can you factor something like this? Here's the question again: Factor $3x^2-11xy+6y^2-xz-4yz-2z^2$.
3
votes
2answers
48 views

Find the value of $\frac{S_{5}S_{2}}{S_{7}}$

If $a$, $b$, $c$ $\in \mathbb R$, we define $S_{k}=\frac{a^k+b^k+c^k}{k}$ (where $k$ is a non-negative integer). Given that $S_{1}=0$, find the value of $$\frac{S_{5}S_{2}}{S_{7}}$$ I tried: ...
0
votes
2answers
56 views

give a complete factored form of the polynomial $-6a^5+48a^4+12a$

Give a complete factored form of the polynomial $-6a^5+48a^4+12a$ I have tried solving this equation and I just cant figure it out. Help me, and give me the answer.
1
vote
3answers
32 views

Factoring when differentiating expressions

I'm having trouble with differentiating a expression. I do it one way, wolfram alpha does it another. Let me show you what I mean. The original expression is this: $$\frac{1}{2u^3}$$ I start by ...
0
votes
4answers
62 views

How can I factor $x^2 + 2\sqrt{3}\,x + 3$? [closed]

$$x^2 + 2\sqrt{3}\,x + 3$$ Anyone could tell me how may I factor this? Thanks a lot
1
vote
1answer
108 views

Factoring $2x^5+13x^4+50x^3+82x^2+56x+13$

Express $2x^5+13x^4+50x^3+82x^2+56x+13$ as a product of five linear factors. The roots of the polynomial may be real or complex. I had to employ the technique of synthetic division iteratively. ...
2
votes
3answers
85 views

Algebra (not so simple) Factoring

I got stuck on this problem from my Math Challenge II Algebra Class: Factorize the following: $$(x^2+xy+y^2)^2-4xy(x^2+y^2)$$ Hint: Let $u=x+y$ and $v=xy$. Here's what I did: ...
2
votes
1answer
63 views

Expanding Square Roots, Why No Negative?

I haven't thought through algebra in a while and the last explanation I received of this seemed arbitrary. I hope I can get some clarification here. I understand that $\sqrt{+a} = \pm b$. Here's ...
0
votes
1answer
25 views

Factoring Polynomial Questions

How do you decide whether to use synthetic division or the factor theorem to help you factor a polynomial? Please help me answer.
0
votes
1answer
19 views

$\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$?

I want to find out if this affermation is true: let $f\in \mathbb{Q}[x]$ such that $\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$? (We know that it has ...
0
votes
2answers
36 views

Cubic factoring question

I'm trying to figure out how a colleague factored an expression. I don't get how: $$a^3+a^2b-(b+1)=(a-1)[a^2+a(b+1)+(b+1)]$$ Multiplying the result I see it's true, but not sure how he got there..is ...
1
vote
1answer
40 views

Factoring and Simplifying

I'm trying to do this problem, $$(4x + 1)^{15}\cdot\frac{1}{3}(12x - 5)^{-\frac{2}{3}}\cdot 12 + (12x - 5)^{\frac{1}{3}}\cdot15(4x + 1)^{14}\cdot 4$$ I've gotten down to, ...
0
votes
3answers
40 views

Basic Algebraic Manipulation

How would I solve for $X$ in this instance? I can't figure out how to get the $X$ variables by themselves and the known values on the other side by themselves. $2(4-X)(4-X)+X = 3$
0
votes
1answer
42 views

Factor this equation [closed]

Can someone factor this for me? $(x^{\frac{n}3}-a^{\frac{n}3})$ I am stuck on it. Let n be any natural number.
0
votes
0answers
31 views

Is this factorization true for all $n$ in the natural numbers

I need to know if $x-a=(x^{\frac{n}3}-a^{\frac{n}3})(x^{\frac{n+1}3}+a^{\frac{n}3} x^{\frac{n}3}+a^{\frac{n+1}3})$ Is true. I know its true for $n=1$, is it true for all natural numbers though?
0
votes
4answers
23 views

What formula do I use for factoring these?

An elementary question, but I am having a lot of discrepancies identifying the correct formula to use, I can do more complex ones but not the simple ones if that makes sense. a) $8x^3 + 1$ b) $m^2 - ...
1
vote
5answers
51 views

Factor fully $98g^2+112g+32$ by decomposition

By looking at this question I understand it is a complex trinomial so do I just decompose it??I have multiplied 98 by 32 getting 3136, but I'm not quite sure what comes next.
-2
votes
2answers
64 views

Factor fully $625-(y-2)^2$

So far, I have used $(y-2)$ twice (multiplying both) because of the exponent being $2$. But, I need to factor and that's when I get confused. Please help!
2
votes
1answer
43 views

Determine 2 values of $k$ so that $36m^2+8m+k$ can be factored over the integers

So, I really need help with this, thank you very much for helping me. Anyway, I understand that $36m^2+8m+k$ is a complex trinomial and when factoring I should use $a^2+2ab+b^2=(a+b)^2$, but this is ...
1
vote
3answers
108 views

how do you factor $x^2 +kx+40$ over the integer

please please help me, I'm having a lot of troubles. I tried to use a^2+2ab+b^2 formula (like i was told) but that's where get lost. I understand that Factoring uses the opposite operation, but 40 ...
2
votes
5answers
110 views

Derivation of factorization of $a^n-b^n$

How does one prove that: $$a^n-b^n=(a-b)\left(a^{n-1}+a^{n-2}b+a^{n-3}b^2+\dots+a^2b^{n-3}+ab^{n-2}+b^{n-1}\right)$$ Better yet, why is $a^n-b^n$ divisible by $a-b$? I would very much appreciate some ...
1
vote
1answer
752 views

Factoring using the 'Criss-Cross' method

my teacher taught our class how to factorize using the criss cross method, and I did not understand what she tried to communicate to the class. The equation I am trying to solve as an example is $7x^2 ...
2
votes
1answer
52 views

What is the solution? Factoring and computing the equation.

If you will be gracious enough to answer, the equation is currently: $$10^x + 15^{x-1}= 20,$$ What is the value of $2x^2$? Please list all steps, if you don't mind. To follow up, what is the name of ...
0
votes
5answers
66 views

Factorization of a degree three polynomial

So I was doing some Vector Calculus homework and was working with Lagrange Multipliers, but then I came across a polynomial that I either forgot how to factor or never learned. I plugged it into ...
1
vote
1answer
65 views

Factorise: $a^4-b^4+c^4-d^4-2(a^2c^2-b^2d^2)+4ac(b^2+d^2)-4bd(a^2+c^2)$

Factorise: $a^4-b^4+c^4-d^4-2(a^2c^2-b^2d^2)+4ac(b^2+d^2)-4bd(a^2+c^2)$. My working: $(a^4-2a^2c^2+c^4)-(b^4-2b^2d^2+d^4)+4ac(b^2+d^2)-4bd(a^2+c^2)$ ...
3
votes
3answers
131 views

Solve $t^4+4 t^3+6 t^2+4 t-32 t^{1/4}+1 = -16 $

I'm trying to solve the following equation: $$(t+1)^4 - 32 t^{\frac{1}{4}}=-16 $$ where t $\geq 0$, which is equivalent to $$t^4+4 t^3+6 t^2+4 t-32 t^{\frac{1}{4}}+1 = -16 $$ Wolfram Alpha tells that ...
2
votes
0answers
73 views

What is the line of thinking to get $[(n^2+3n+1)^2-5n(n+1)^2]$ from $(n^4+n^3+n^2+n+1)$?

There is this one little tiny step along the working that I don't quite understand, but I think it is better if I write the whole problem and solution for clarity. Problem: Factorise $5^{1995}-1$ ...
0
votes
1answer
68 views

Substitution to linear + nth power form

Given an arbitrary polynomial: $$a_0 + a_1x + a_2x^2 ... a_nx^n$$ Does there exist a series of substitutions (or single substitution if you choose to combine them) that leaves this function in the ...
1
vote
6answers
164 views

How to factorize $2x^2+5x+3$?

I'm doing pre-calculus course at coursera.org and I'm in trouble with this solution $$2x^2 +5x +3 = (2x+3)(x+1)$$ By trial, using ac-method I got stuck: $$ ac = (2)(3) = 6\\ 6 + ? = 5 \Rightarrow~ ? ...
0
votes
4answers
58 views

Factoring with rational exponents

I'm not quite sure how to do this question. Every way that I tried doing it didn't yield an answer that is equivalent to the original question. $$(2x+1)^{2/3}-4(2x+1)^{-1/3}$$ When I tried doing ...
2
votes
4answers
120 views

How to determine if $2+x+y$ is a factor of $4-(x+y)^2$?

I know it is a factor but how could have I determined that it was? Feel free to link whatever concept is needed than solve it. Studying for clep and it's one of the practice problems. When I expand it ...
1
vote
2answers
75 views

Question about Polynomial Factor Theorem

I was reading the solution to an algebra problem but got stuck at one part. Problem is here: (http://math.la.asu.edu/~ifulman/mat194/problem-solving.pdf) Example 4.2.6 -- page 140 of the PDF (the book ...
0
votes
3answers
120 views

Prove that $ x^n - y^n = (x-y) (x^{n-1}+x^{n-2}y\,+ \,\,…\,\,+ y^{n-1})$ [closed]

Prove that $ x^n - y^n = (x-y). (x^{n-1}+x^{n-2}y\,+ \,\,...\,\,+ y^{n-1}) $; $\,\,\,\,\,$$x,y \in \mathbb{R}$
2
votes
2answers
33 views

Polynomial With Imaginary Roots

Working on question 1 here http://www.sosmath.com/cyberexam/precalc/EA2002/EA2002.html Find a polynomial with integer coefficients that has the following zeros: ...
0
votes
3answers
55 views

Can someone please explain how this was factored?

How was $x^2(x+1)-4(x+1)$ factored into $$(x^2-4)(x+1)?$$ I know this seems very basic but can someone please explain this?
2
votes
1answer
55 views

Showing $(a+b+c)(x+y+z)=ax+by+cz$ given other facts

$$x^2-yz/a=y^2-zx/b=z^2-xy/c$$ None of these fractions are equal to 0.We need to show that, $(a+b+c)(x+y+z)=ax+by+cz$ This question comes from a chapter that wholly deals with factoring ...
0
votes
1answer
61 views

Please help me with factorisation

Is it possible to write $$64x^6-112x^4+56x^2-7$$ in linear factors? If so, what are they? (Finding it really difficult to ask this question!!)
2
votes
1answer
175 views

Factorizing $(x-1)(x-3)(x-5)(x-7)-64$

We need to factorize: $$(x-1)(x-3)(x-5)(x-7)-64$$ We can, by the rational root theorem, see that there are no roots of this polynomial.Next observation is that $64=(8)^2$. So this means that if the ...
0
votes
1answer
94 views

How long is an arrow in the air?

The height $h$ of an arrow in feet is modeled by $h(t) = -16t^2 + 63t + 4$, where $t$ is the time in seconds since the arrow was shot. How long is the arrow in the air? Could someone explain where to ...
2
votes
3answers
136 views

Factoring out an exponential?

I have the following expression $$\frac{2^{k+1}(k+1)!}{(k+1)^{k+1}}\cdot\frac{k^k}{2^k k!}$$ I get $$\frac{2(k+1)(k^k)}{(k+1)^{k+1}}$$ But how do I factor out the ${(k+1)}^{k+1}$
1
vote
3answers
69 views

How do you factor $(10x+24)^2-x^4$?

I tried expanding then decomposition but couldn't find a common factor between two terms
1
vote
2answers
120 views

Factoring Complex Trinomials

What is the answer for factoring: $$10r^2 - 31r + 15$$ I have tried to solve it. This was my prior attempt: $$10r^2 - 31r + 15\\ = (10r^2 - 25r) (-6r + 15)\\ = -5r(-2r+5) -3 (2r-5) $$
0
votes
1answer
57 views

Polynomial (third degree)

A third degree polynomial $p(x)=0$ when $x=1$ and $x=3$. We also learn that $p(x) \geq 0 $ when $x \geq 1$ and $p(2) =2$. Determine $p(x)$. How should I proceed? I presume no calculus is needed.
1
vote
0answers
38 views

Given a cubic $f(x)$ with specified negative real roots $-a,-b,-c$, what happens when we search for solutions to $f(x)=d$?

Noting Roots of a Certain type of Cubic Equation, what if we have the following simpler form for real $d$: $$(x+a)(x+b)(x+c)=d\tag{1}$$ (With $a,b,c\in \mathbb R^+$.) Clearly, depending on $d$, the ...
1
vote
1answer
90 views

Why does completing the square give you the minimum point?

Say we have an equation:$y=$ ${x^2} + 2x + 1$ Completing the square we get: $\eqalign{ & y={x^2} + 2x + 1 \cr & = {(x + 1)^2} - {(1)^2} + 1 \cr & = {(x + 1)^2} \cr} $ The ...
3
votes
2answers
61 views

How do you factor this? $x^3 + x - 2$

How do you factor $x^3 + x - 2$? $(x-1)(x^2 + x + 2)$ Note the factored form here. Thanks!
-2
votes
3answers
126 views

How do I factor this polynomial $x^5-4x^3+8x^2-32$? [closed]

How do I factor this polynomial? $p(x) = x^5-4x^3+8x^2-32$
2
votes
3answers
108 views

Factor $(x+y)^7-(x^7+y^7)$

So I was doing some practice problems to prepare upcoming math contests. This is one of the problems: Factor $(x+y)^7-(x^7+y^7)$ I got zero for $(x+y)^7-(x^7+y^7)$, however, the solutions ...