Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Please check my answer in factoring this equations:

Question 1. Factor $(x+1)^4+(x+3)^4-272$.

Solution: $$\begin{eqnarray}&=&(x+1)^4+(x+3)^4-272\\&=&(x+1)^4+(x+3)^4-272+16-16\\ &=&(x+1)^4+(x+3)^4-256-16\\ &=&\left[(x+1)^4-16\right]+\left[(x+3)^4-256\right]\\ &=&\left[(x+1)^2+4\right]\left[(x+1)^2-4\right]+\left[(x+3)^2+16\right]\left[(x+3)^2-16\right]\\ &=&\left[(x+1)^2+4\right]\left[(x+1)^2-4\right]+\left[(x+3)^2+16\right]\left[(x+3)-4\right]\left[(x+3)+4\right]\end{eqnarray}.$$

Question 2. Factor $x^4+(x+y)^4+y^4$

Solution: $$\begin{eqnarray}&=&(x^4+y^4)+(x+y)^4\\ &=&(x^4+y^4)+(x+y)^4+2x^2y^2-2x^2y^2\\ &=&(x^4+2x^2y^2+y^4)+(x+y)^4-2x^2y^2\\ &=&(x^2+y^2)^2+(x+y)^4-2x^2y^2 \end{eqnarray}$$

I am stuck in question number 2, I dont know what is next after that line.

share|cite|improve this question

3 Answers 3

up vote 1 down vote accepted

\begin{equation} \begin{split} \ & x^4+y^4+(x+y)^4\\ \ =& (x^2+y^2)^2-2x^2y^2+(x^2+y^2+2xy)^2\\ \ =& (x^2+y^2)^2-2x^2y^2+(x^2+y^2)^2+4xy(x^2+y^2)+4x^2y^2\\ \ =& 2((x^2+y^2)^2+x^2y^2+2xy(x^2+y^2))\\ \ =& 2(x^2+y^2+xy)^2 \end{split} \end{equation}

share|cite|improve this answer
Thank you Samrat Mukhopadhyay! – Al-Ahmadgaid Asaad Jul 7 '13 at 10:57
welcome! @al-ahmadgaid-asaad – Samrat Mukhopadhyay Jul 7 '13 at 11:08

\begin{align*} (x+y)^4+x^4+y^4&=2(x^4+2x^3y+3x^2y^2+2xy^3+y^4)\\ &=2(x^4+2x^3y+2x^2y^2+x^2 y^2+2 x y^3+y^4)\\ &=2(x^4+2(xy+y^2)x^2+(xy+y^2)^2)\\ &=2(x^2+xy+y^2)^2 \end{align*}

share|cite|improve this answer
Thank you yanbo! – Al-Ahmadgaid Asaad Jul 7 '13 at 10:57
You are very welcome! – BlackAdder Jul 7 '13 at 11:00

For the first, I will put $y=\frac{x+1+x+3}2=x+2$

so that $x +1=y-1, x+3=y+1$ and the odd powers of $y$ vanish in $(y-1)^4+(y+1)^4$

$$\implies (x+1)^4+(x+3)^4-272=(y-1)^4+(y+1)^4-272$$


$$=2\{y^4+(15-9)y^2-135\}=2(y^2+15)(y^2-9) =2(y^2-15)(y+3)(y-3)$$


$$\text{Now, }(x+2)^2-15=x^2+4x+4-15=x^2+4x-11\text{ which is not reducible}$$

share|cite|improve this answer
Thank you lab bhattacharjee! – Al-Ahmadgaid Asaad Jul 7 '13 at 10:58
@Al-AhmadgaidAsaad, my pleasure – lab bhattacharjee Jul 7 '13 at 11:39

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.