`@` Angelmare

Đáp án:

a)

`P(x)=3x^2+7+2x^4-3x^2-4-5x+2x^3`

`P(x)=2x^4+2x^3+(3x^2-3x^2)-5x+(7-4)`

`P(x)=2x^4+2x^3-5x+3`

`-` Bậc: `4`

`-` Hệ số tự do: `3`

`-` Hệ số cao nhất: `2`

`Q(x)=-3x^3+2x^2-x^4+x+x^3+4x-2+5x^4`

`Q(x)=(5x^4-x^4)-(3x^3-x^3)+2x^2+(x+4x)-2`

`Q(x)=4x^4-2x^3+2x^2+5x-2`

`-` Bậc: `4`

`-` Hệ số tự do: `-2`

`-` Hệ số cao nhất: `4`

b) Thay x=-1 vào P(x) ta có:

`P(-1)=2*(-1)^4+2*(-1)^3+5*1+3=8`

Thay x=0 vào Q(x) ta có:

`Q(0)=4*0^4-2*0^3+2*0^2+5*0-2=-2`

c) `G(x)=P(x)+Q(x)`

`G(x)=(2x^4+2x^3-5x+3)+(4x^4-2x^3+2x^2+5x-2)`

`G(x)=2x^4+2x^3-5x+3+4x^4-2x^3+2x^2+5x-2`

`G(x)=(2x^4+4x^4)+(2x^3-2x^3)+2x^2-(5x-5x)+(3-2)`

`G(x)=6x^4+2x^2+1`

d) `P(x)=A(x)+Q(x)`

`to A(x)=P(x)-Q(x)`

`A(x)=(2x^4+2x^3-5x+3)-(4x^4-2x^3+2x^2+5x-2)`

`A(x)=2x^4+2x^3-5x+3-4x^4+2x^3-2x^2-5x+2`

`A(x)=(2x^4-4x^4)+(2x^3+2x^3)-2x^2-(5x+5x)+(3+2)`

`A(x)=-2x^4+4x^3-2x^2-10x+5`

e) T/c:

`6x^4;2x^2 ge 0 AA x`

`to G(x)=6x^4+2x^2+1 ge 0+0+1=1 gt 0 AA x`

`to G(x)` luôn dương (đpcm)

`M(x)-P(x)+Q(x)=x^2-1`

`M(x)=x^2-1+(2x^4+2x^3-5x+3)-(4x^4-2x^3+2x^2+5x-2)`

`M(x)=x^2-1+2x^4+2x^3-5x+3-4x^4+2x^3-2x^2-5x+2`

`M(x)=(2x^4-4x^4)+(2x^3+2x^3)+(x^2-2x^2)-(5x+5x)+(3+2-1)`

`M(x)-2x^4+4x^3-x^2-10x+4`