factorize--prophetes.ai

factorize

factorize, v. (ˈfæktəraɪz) [f. as factorial a.1 + -ize.] 1. trans. (U.S. Law.) In Vermont and Connecticut, = garnish.1864 in Webster. 1878 [see factor n. 5 c.]. 2. Math. a. To break up (a quantity) into factors.1886 G. Chrystal Algebra I. vii. 133 The decomposition of x2 + y2 + xy = (x + y + √xy ) (... Oxford English Dictionary

prophetes.ai

factorize

factorizefactorise, / ˈfæktəraɪz; `fæktəˌraɪz/ v [Tn](mathematics 数) find the factors of (a number) 分解（某数的）因数. 牛津英汉双解词典

prophetes.ai

Factorize x^3+3 Task: Factorize $x^3+3$ in $\mathbb{R}$ and $GF(7)$ I think the solution is $x^3+3$ in both cases, so the polynomial already is irreductible. Is my assuption is right, how do i show that?

To check whether a cubic polynomial is irreducible over a given field, it is sufficient to check whether $f$ has any roots in that field (why?). In $GF(7)$ this is easy: there are only $7$ possible roots, so you can simply evaluate $f(x)$ for all $x \in GF(7)$. Over $\mathbb{R}$, remember that every...

prophetes.ai

Factorize $8x^3 + 12x^2 -2x -3$ How do I factorize this - $$8x^3 + 12x^2 -2x -3$$ I tried splitting the middle term but that didn't work , I tried factor theorem with various factors but even that didn't work. What ca...

One possibility is to note the $8x^3$ and $3$ at each end and then, thinking of the rational root theorem, see if any of $x=\pm3,\pm1,\pm\frac32,\pm\frac12,\pm\frac34,\pm\frac14,\pm\frac38,\pm\frac18$ make your expression zero. Three of them do in this case, namely $x=-\frac32,+\frac12 \text{ and } ...

prophetes.ai

Factorize $a^2-ab-bc\pm c^2$ I got this question in a test but it did not specify the variable with respect to which I was supposed to factorize $$a^2-ab-bc\pm c^2$$ where it could be just $a(a-b)-c(b\pm c)$ but no ...

We observe that \begin{eqnarray} a^2-ab-bc+c^2&=&a^2+c^2-b(a+c)\\\ &=&(a+c)^2-b(a+c)-2ac\\\ &=&(a+c)^2-b(a+c)+\frac{b^2}{4}-\frac{b^2+8ac}{4}\\\ &=&\left(a+c-\frac{b}{2}\right)^2-\frac{b^2+8ac}{4}. \end{eqnarray} Hence, if $b^2+8ac\geq 0$ then $$ a^2-ab-bc+c^2=\left(a+c-\frac{b}{2}+\sqrt{\frac{b^2+8...

prophetes.ai

Factorize a number into coprime numbers I want to know if there is a way to factorize a number into coprime numbers; for example $N = a_1 \cdot a_2 \cdot a_3 \cdots a_i$ And $a_i$ and $a_j$ are coprime for any $i \n...

You can construct this from the prime factorization. Just group all the appearances of the same prime factor to obtain $N=\prod p_i^{n_i}$. Since $p_i,p_j$ are distinct primes, any power of them has no non-unit common factors.

prophetes.ai

Need to factorize$ x^7 - x$ to irreducible polynomials over $\operatorname{GF}(2)$ Can some one help me with an easy method to solve this question? > Factorize $x^7 - x$ to irreducible polynomials over $\operatornam...

Using that modulo $\;2\;$ we have $\;(a+b)^2=a^2+b^2\;$ : $$x^7-x=x(x^6-1)=x(x^3-1)^2=x(x-1)^2(x^2+x+1)^2$$ Of course, $\;-1=1\;$ so you can write $\;x-1\;$ or $\;x+1\;$ . It's the same.

prophetes.ai

How to factorize this quadratic? How do i factorize this equation: $a(b-c)x^2 + b(c-a)x + c(b-a) = 0$ I tried the quadratic formula, but the discriminant is not factorising into a perfect square. Please help!

HINT: Observe that $$a(b-c)-c(b-a)=-b(c-a)$$ Put the value of $b(c-a),$ and take out common from the first two terms & the last two terms and see what happens?

prophetes.ai

Factorize the polynomial $x^3+y^3+z^3-3xyz$ I want to factorize the polynomial $x^3+y^3+z^3-3xyz$. Using Mathematica I find that it equals $(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$. But how can I factorize it by hand?

Use Newton's identities: $p_3=e_1 p_2 - e_2 p_1 + 3e_3$ and so $p_3-3e_3 =e_1 p_2 - e_2 p_1 = p_1(p_2-e_2)$ as required. Here $p_1= x+y+z = e_1$ $p_2= x^2+y^2+z^2$ $p_3= x^3+y^3+z^3$ $e_2 = xy + xz + yz$ $e_3 = xyz$

prophetes.ai

Factorising $X^n+...+X+1$ in $\mathbb{R}$ How can factorize this polynom in $\mathbb{R}$: $X^n+...+X+1$ I already try to factorize it in $\mathbb{C}$ but I couldn't find a way to turn to $\mathbb{R}$

We have $$\sum_{k=0}^n x^k=\frac{x^{n+1}-1}{x-1}$$ hence $$\sum_{k=0}^n x^k=\prod_{k=1}^{n}\left(x-e^{2ik\pi/{n+1}}\right)$$ so if $n$ is odd say $n=2p+1$ then $$\sum_{k=0}^{2p+1}=\prod_{k=1}^{2p+1}\left(x-e^{2ik\pi/{2p+2}}\right)=(x+1)\prod_{k=1}^{p}\left(x-e^{2ik\pi/{2p+2}}\right)\prod_{k=p+2}^{2p...

prophetes.ai

Factorize a third degree polynomial I'm currently trying to solve a problem which asks if a 3x3 matrix is diagonalizable, I know the method but when it comes to finding the roots, I have a third degree polynomial and ...

Try to "guess" some rational root $\;\cfrac rs\;$ , which by the Rational Root Theorem must fulfill $\;r\,\mid\,-2\;,\;\;s\,\mid\,1\;$ , and indeed $\;2\;$ is a root, so divide by $\;x-2\;$ : $$x^3-3x-2=(x-2)(x^2+2x+1)=(x-2)(x+1)^2$$ and you have one simple root and one double one. If there is no ra...

prophetes.ai

How to factorize $(x-2)^5+x-1$? This is a difficult problem. How to factorize this? $$(x-2)^5+x-1$$ we can't do any thing now and we should expand it first: $$x^5-10x^4+40x^3-80x^2+81x-33$$ but I can't factorize it.

Using the substitution $t=x-2$, it suffices to factor $t^5 + t+1$. But $t^2+t+1$ divides $t^5+t+1$. Why? $t^2+t+1 = (t-\zeta)(t-\zeta')$, where $\zeta,\zeta'$ are the primitive $3rd$ roots of unity, and thus $\zeta^5 + \zeta + 1 = \zeta^2 + \zeta + 1 = 0$, similarly for $\zeta'$. Now, the quotient $...

prophetes.ai

equation representing 2 straight lines Let us assume this equation is given to us we have to factorize it $$12x^2 +7xy-10y^2+13x+45y-3=0$$ By solving we get that this represents two straight lines. But how to factoriz...

First factorize $12x^2+7xy-10y^2$ as the product of two factors $(ax+by)(cx+dy)$. It is just like factoring an ordinary quadratic.

prophetes.ai

How to factorize $n(n+1)(n+2)(n+3)+1$? How to factorize $n(n+1)(n+2)(n+3)+1$ ? It's turned into the lowest power of $n$ ever possible but the question wants me to factorize it. How can I do that ?

$$n(n+3)=n^2+3n$$ and $$(n+1)(n+2)=n^2+3n+2$$ $$\implies n(n+1)(n+2)(n+3)=(n^2+3n)[(n^2+3n)+2]+1=[(n^2+3n)+1]^2$$

prophetes.ai

Does this polynomial factorize further? I just did a national exam and this question was in it; I am convinced this does not work: > Given that $(x - 1)$ is a factor of $x^3 + 3x^2 + x - 5$, factorize this cubic full...

The discriminant of your polynomial is $$\Delta=4^2-4\cdot 5<0$$ so this has no real roots, that is, factorization over $\Bbb R$ is finished. On the other hand, factorization over $\Bbb C$ is possible, yielding $$x^2+4x+5=(x+2)^2+1=(x+2-i)(x+2+i)$$

prophetes.ai

factorize

Answers

MindMap

Sources

factorize

factorize

Factorize x^3+3 Task: Factorize $x^3+3$ in $\mathbb{R}$ and $GF(7)$ I think the solution is $x^3+3$ in both cases, so the polynomial already is irreductible. Is my assuption is right, how do i show that?

Factorize $8x^3 + 12x^2 -2x -3$ How do I factorize this - $$8x^3 + 12x^2 -2x -3$$ I tried splitting the middle term but that didn't work , I tried factor theorem with various factors but even that didn't work. What ca...

Factorize $a^2-ab-bc\pm c^2$ I got this question in a test but it did not specify the variable with respect to which I was supposed to factorize $$a^2-ab-bc\pm c^2$$ where it could be just $a(a-b)-c(b\pm c)$ but no ...

Factorize a number into coprime numbers I want to know if there is a way to factorize a number into coprime numbers; for example $N = a_1 \cdot a_2 \cdot a_3 \cdots a_i$ And $a_i$ and $a_j$ are coprime for any $i \n...

Need to factorize$ x^7 - x$ to irreducible polynomials over $\operatorname{GF}(2)$ Can some one help me with an easy method to solve this question? > Factorize $x^7 - x$ to irreducible polynomials over $\operatornam...

How to factorize this quadratic? How do i factorize this equation: $a(b-c)x^2 + b(c-a)x + c(b-a) = 0$ I tried the quadratic formula, but the discriminant is not factorising into a perfect square. Please help!

Factorize the polynomial $x^3+y^3+z^3-3xyz$ I want to factorize the polynomial $x^3+y^3+z^3-3xyz$. Using Mathematica I find that it equals $(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$. But how can I factorize it by hand?

Factorising $X^n+...+X+1$ in $\mathbb{R}$ How can factorize this polynom in $\mathbb{R}$: $X^n+...+X+1$ I already try to factorize it in $\mathbb{C}$ but I couldn't find a way to turn to $\mathbb{R}$

Factorize a third degree polynomial I'm currently trying to solve a problem which asks if a 3x3 matrix is diagonalizable, I know the method but when it comes to finding the roots, I have a third degree polynomial and ...

How to factorize $(x-2)^5+x-1$? This is a difficult problem. How to factorize this? $$(x-2)^5+x-1$$ we can't do any thing now and we should expand it first: $$x^5-10x^4+40x^3-80x^2+81x-33$$ but I can't factorize it.

equation representing 2 straight lines Let us assume this equation is given to us we have to factorize it $$12x^2 +7xy-10y^2+13x+45y-3=0$$ By solving we get that this represents two straight lines. But how to factoriz...

How to factorize $n(n+1)(n+2)(n+3)+1$? How to factorize $n(n+1)(n+2)(n+3)+1$ ? It's turned into the lowest power of $n$ ever possible but the question wants me to factorize it. How can I do that ?

Does this polynomial factorize further? I just did a national exam and this question was in it; I am convinced this does not work: > Given that $(x - 1)$ is a factor of $x^3 + 3x^2 + x - 5$, factorize this cubic full...