Question

Factor x^10 +y^4 using the sum of squares (hint: x^10 = (x^5)^2 and y^4 = (y^2)^2)

Answer 1

$x^{10} +y^4$ cannot be factorized that is cannot be represented as products of lower degree polynomials.

Solution:

Need to factor $x^{10} +y^4$ using the sum of squares that is we need to factorize $x^{10} +y^4$

$x^{10}+y^{4}=x^{(5 \times 2)}+y^{(2 \times 2)}$

Using law of exponent $\mathrm{a}^{(\mathrm{m} \times \mathrm{n})}=\left(\mathrm{a}^{\mathrm{m}}\right)^{\mathrm{n}}$

On applying this we get,

$x^{10}+y^{4}=\left(x^{5}\right)^{2}+\left(y^{2}\right)^{2}$

But there is no direct formula of $\mathrm{a}^{2}+\mathrm{b}^{2}$ which can provide factors.

Hence we can say that $x^{10} +y^4$ cannot be factorize that is cannot be represented as products of lower degree polynomials.