Question

Find the degree of the polynomial

${(x+(x^3-1{)}^{\frac{1}{2}})}^5+{(x-(x^3-1{)}^{\frac{1}{2}})}^5$

Answer 1

Recall that,

$(x+y{)}^n=n_{c_0}{x}^n+n_{c_1}{x}^{n-1}y+n_{c_2}{x}^{n-2}{y}^2+n_{c_3}{x}^{n-3}{y}^3+.....+n_{c_n}{y}^n$

Given polynomial is,

${(x+(x^3-1{)}^{\frac{1}{2}})}^5+{(x-(x^3-1{)}^{\frac{1}{2}})}^5$

$={}^5{C}_0.x^5+{}^5{C}_1{x}^4.(x^3-1{)}^{\frac{1}{2}}+{}^5{C}_2{x}^3(x^3-1)+{}^5{C}_3{x}^3(x^3-1{)}^{\frac{3}{2}}+{}^5{C}_{4}x(x^3-1{)}^2+{}^5{C}_5{x}^5(x^3-1{)}^{\frac{5}{2}}+{}^5{C}_0{x}^5-{}^5{C}_1{x}^4.(x^3-1{)}^{\frac{1}{2}}+{}^5{C}_2{x}^3(x^3-1)+{}^5{C}_3{x}^2(x^3-1{)}^{\frac{3}{2}}+{}^5{C}_{4}x(x^3+1{)}^2-{}^5{C}_{3}x(x^3-1{)}^{\frac{5}{2}}$

$={}^5{C}_0{x}^5+{}^5{C}_2{x}^6-x^3+{}^5{C}_4{x}^7+x-2x^4+{}^5{C}_0{x}^5+{}^5{C}_2{x}^6-x^3+{}^5{C}_4{x}^7+x-2x^4$

$=2({}^5{C}_0{x}^5+{}^5{C}_2{x}^6-x^3+{}^5{C}_4{x}^7+x-2x^4)$

Here, highest power is $7$.

Therefore, degree of the polynomial is $7$.