Question 79
a. Let $f(x) = x^2 + 8$. Prove that $f(x\sqrt{2})$ is the product $P(x)Q(x)$ of two polynomials of degree two with integer coefficients.
b. Let $g(x) = x^4 + x^3 + x^2 + x + 1$. Prove that $g(5x^2)$ is the product $R(x)S(x)$ of two polynomials of degree four with integer coefficients.
c. Let $h(x) = (x+1)^{13} - x^{13} - 1$. Prove that $h(x)$ is the product $T(x)U(x)$ of two polynomials of degree six with integer coefficients.
b. Let $g(x) = x^4 + x^3 + x^2 + x + 1$. Prove that $g(5x^2)$ is the product $R(x)S(x)$ of two polynomials of degree four with integer coefficients.
c. Let $h(x) = (x+1)^{13} - x^{13} - 1$. Prove that $h(x)$ is the product $T(x)U(x)$ of two polynomials of degree six with integer coefficients.