a,f(x)+g(x)=\(\left(a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0\right)+\left(b_nx^{n-1}+...+b_1x+b_0\right)\)

=\(a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0+b_nx^n+b_{n-1}x^{n-1+...+b_1x+b_0}\)

\(=\left(a_nx^n+b_nx^n\right)+\left(a_{n-1}x^{n-1}+b_{n-1}x^{n-1}\right)+...+\left(a_1x+b_1x\right)+\left(a_0+b_0\right)\)

b

f(x)+g(x)=\(\left(a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0\right)+\left(b_nx^n+b_{n-1}x^{n-1}+...+b_1x+b_0\right)\)

\(=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0-b_nx^n-b_{n-1}-b_1x+b_0\)

\(=(a_nx^n-b_nx^n)+(a_{n-1}x^{n-1}-b_{n-1}x^{n-1})+...+(a_1x-b_1x)+\left(a_0+b_0\right)\)

\(=\left(a_n-b_n\right)x^n+(a_{n-1}-b_{n-1})x^{n-1}+...+\left(a_1-b_1\right)x+\left(a_0-b_0\right)\)

a. Ta có: f(x) + h(x) = g(x)

Suy ra: h(x) = g(x) – f(x) = (x4 – x3 + x2 + 5) – (x4 – 3x2 + x – 1)

= x4 – x3 + x2 + 5 – x4 + 3x2 – x + 1

= -x3 + 4x2 – x + 6

b. Ta có: f(x) – h(x) = g(x)

Suy ra: h(x) = f(x) – g(x) = (x4 – 3x2 + x – 1) – (x4 – x3 + x2 + 5)

= x4 – 3x2 + x – 1 – x4 + x3 – x2 – 5

= x3 – 4x2 + x – 6

Gỉa sử P(x) có một nghiệm nguyên là \(x_0\left(x_0\ne0\right)\)

Ta có \(P\left(x\right)=a_nx_0^n+a_{n-1}x_0^{n-1}+...+a_1x_0+a_0=0.\)

Như vậy \(P\left(x_0\right)=0⋮x_0\)và các số hạng \(a_nx_0^n+a_{n-1}x_0^{n-1}+...+a_1x_0\)đều chia hết cho \(x_0\), suy ra \(a_0\)cũng phải chia hết \(x_0\)tức \(x_0\)là ước của \(a_0\)

\(f\left(1\right)=a_{2017}+a_{2016}+...+a_3+a_2+a_1+a_0\)

\(f\left(-1\right)=-a_{2017}+a_{2016}+...-a_3+a_2-a_1+a_0\)

\(f\left(1\right)+f\left(-1\right)=2\left(a_{2016}+a_{2014}+...+a_2+a_0\right)\)

\(S=\frac{f\left(1\right)+f\left(-1\right)}{2}=\frac{3^{2017}+1}{2}\)