Factorize the polynomial ${x}^{3}+{y}^{3}+{z}^{3}-3xyz$

Holetaug
2022-07-13
Answered

Factorize the polynomial ${x}^{3}+{y}^{3}+{z}^{3}-3xyz$

You can still ask an expert for help

Monserrat Cole

Answered 2022-07-14
Author has **12** answers

Solution:

$\begin{array}{r}{x}^{3}+{y}^{3}+{z}^{3}-3xyz\\ & ={x}^{3}+{y}^{3}+3{x}^{2}y+3x{y}^{2}+{z}^{3}-3xyz-3{x}^{2}y-3x{y}^{2}\\ & =(x+y{)}^{3}+{z}^{3}-3xy(x+y+z)\\ & =(x+y+z)((x+y{)}^{2}+{z}^{2}-(x+y)z)-3xy(x+y+z)\\ & =(x+y+z)({x}^{2}+2xy+{y}^{2}+{z}^{2}-yz-xz-3xy)\\ & =(x+y+z)({x}^{2}+{y}^{2}+{z}^{2}-xy-yz-zx)\end{array}$

$\begin{array}{r}{x}^{3}+{y}^{3}+{z}^{3}-3xyz\\ & ={x}^{3}+{y}^{3}+3{x}^{2}y+3x{y}^{2}+{z}^{3}-3xyz-3{x}^{2}y-3x{y}^{2}\\ & =(x+y{)}^{3}+{z}^{3}-3xy(x+y+z)\\ & =(x+y+z)((x+y{)}^{2}+{z}^{2}-(x+y)z)-3xy(x+y+z)\\ & =(x+y+z)({x}^{2}+2xy+{y}^{2}+{z}^{2}-yz-xz-3xy)\\ & =(x+y+z)({x}^{2}+{y}^{2}+{z}^{2}-xy-yz-zx)\end{array}$

asked 2021-02-25

Find a polynomial of the specified degree that has the given zeros. Degree 4, zeros -2, 0, 2, 4

asked 2020-10-28

Find a polynomial with integer coefficients that satisfies the given conditions. Q has degree 3 and zeros 0 and i.

asked 2022-06-22

Prove that ${t}_{n}=1+\frac{\mathrm{ln}(2n)}{2n}+o\left(\frac{1}{n}\right)$ as n tends to infinity.

asked 2022-03-31

I have no idea how this equation:

${({x}^{2}+{y}^{2}-1)}^{3}-{x}^{2}{y}^{3}=0$

asked 2021-09-06

Determine which functions are polynomial functions. For those that are, identify the degree.

$f\left(x\right)={x}^{\frac{1}{2}}-3{x}^{2}+5$

asked 2021-09-07

Factor each of the following polynomials as the product of two polynomials of degree 1 in ${Z}_{10}\left[x\right]$ .

$x+9$

asked 2022-04-04

Find a ' number p so that $f=3{x}^{3}+2{x}^{2}-5x+1$ is divided by $x-2$ in $\mathbb{Z}}_{p$