a: \(A\left(x\right)=2x^4-x^3+3x^2+9x-2\)

\(B\left(x\right)=2x^4-5x^3-x+9\)

\(C\left(x\right)=x^4+4x^2+5\)

A(x): bậc 4; hệ số cao nhất là 2; hệ số tự do là -2

B(x): bậc 4; hệ số cao nhất là 4; hệ số tự do là 9

b: M(x)=A(x)+B(x)=4x^4-6x^3+3x^2+8x+7

N(x)=B(x)-A(x)=-4x^3-3x^2-10x+11

c: Q(x)=-N(x)=4x^3+3x^2+10x-11

a)\(P\left(x\right)=x^4+3\)

b)\(Q\left(x\right)=-x^3-2x^2-14x-1\)

`@` `\text {Ans}`

`\downarrow`

`a)`

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

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

`= 2x^4 + 2x^3 - 5x + 3`

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

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

`= 4x^4 + 4x^3 + 2x^2 + 5x - 2`

`b)`

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

`= 2*1 + 2*(-1) + 5 + 3`

`= 2 - 2 + 5 + 3`

`= 8`

___

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

`= 4*0 + 4*0 + 2*0 + 5*0 - 2`

`= -2`

`c)`

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

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

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

`= 6x^4 + 6x^3 + 2x^2 + 1`

`d)`

`G(x) = 6x^4 + 6x^3 + 2x^2 + 1`

Vì `x^4 \ge 0 AA x`

    `x^2 \ge 0 AA x`

`=> 6x^4 + 2x^2 \ge 0 AA x`

`=> 6x^4 + 6x^3 + 2x^2 + 1 \ge 0`

`=> G(x)` luôn dương `AA` `x`

a: \(P\left(x\right)=-5x^4+2x^2-8x+\dfrac{1}{2}\)

\(Q\left(x\right)=4x^4+2x^3-5x^2-6x+\dfrac{3}{2}\)

b: \(A\left(x\right)=-5x^4+2x^2-8x+\dfrac{1}{2}+4x^4+2x^3-5x^2-6x+\dfrac{3}{2}=-x^4+2x^3-3x^2-14x+2\)

\(B\left(x\right)=-5x^4+2x^2-8x+\dfrac{1}{2}-4x^4-2x^3+5x^2+6x-\dfrac{3}{2}=-9x^4-2x^3+7x^2-2x-1\)

a)\(Q\left(x\right)=2x^3+4x^4-6x-5x^2+\dfrac{3}{2}\)

\(P\left(x\right)=2x^2-5x^4-8x+\dfrac{1}{2}\)

\(A\left(x\right)=2x^3-x^4-3x^2+2-14x\)

\(B\left(x\right)=-2x^3-9x^4-2x+7x^2-1\)

\(P\left(x\right)=3x^2+7+2x^4-3x^2-4-5x+2x^3\)

\(=2x^4+2x^3+\left(3x^2-3x^2\right)-5x-4+7\)

\(=2x^4+2x^3-5x+3\)

\(Q\left(x\right)=-3x^3+2x^2-x^4+x+x^3+4x-2+5x^4\)

\(=\left(5x^4-x^4\right)+\left(-3x^3+x^3\right)+2x^2+\left(x+4x\right)-2\)

\(=4x^4-2x^3+2x^2+5x-2\)

a: \(P\left(x\right)=5x^5-4x^4-2x^3+4x^2+3x+6\)

Bậc là 5

\(Q\left(x\right)=-5x^5+4x^4+2x^3-4x^2+7x+\dfrac{1}{4}\)

Bậc là 5

b: H(x)=P(x)+Q(x)

\(=5x^5-4x^4-2x^3+4x^2+3x+6-5x^5+4x^4+2x^3-4x^2+7x+\dfrac{1}{4}\)

=10x+6,25

c: Để H(x)=0 thì 10x+6,25=0

hay x=-0,625

\(b,N\left(x\right)=\left(4x^5-4x^5\right)+x^4-2x^3-3x^2+\left(3x-5x\right)+1\\ =x^4-2x^3-3x^2-2x\\ c,P\left(x\right)=\left(-2x^6+2x^6\right)+x^5+\left(-2x^4+2x^4\right)+2x^3-x+5\\ =x^5+2x^3-x+5\)

`7,`

`a,`

\(M(x) = - 5x ^ 4 + 3x ^ 5 + x(x ^ 2 + 5) + 14x ^ 4 - 6x ^ 5 - x ^ 3 + x - 1 \)

\(M(x)=-5x^4+3x^5+x^3+5x+14x^4-6x^5-x^3+x-1\)

`M(x)=(3x^5-6x^5)+(-5x^4+14x^4)+(x^3-x^3)+(5x+x)-1`

`M(x)=-3x^5+9x^4+6x-1`

\(N(x)=x ^ 4 (x - 5) - 3x ^ 3 + 3x + 2x ^ 5 - 4x ^ 4 + 3x ^ 3 - 5 \)

\(N(x)=x^5-5x^4-3x^3+3x+2x^5-4x^4+3x^3-5\)

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

`N(x)=3x^5-9x^4+3x-5`

`b,`

`H(x)=M(x)+N(x)`

\(H(x)=(-3x^5+9x^4+6x-1)+(3x^5-9x^4+3x-5) \)

`H(x)=-3x^5+9x^4+6x-1+3x^5-9x^4+3x-5`

`H(x)=(-3x^5+3x^5)+(9x^4-9x^4)+(6x+3x)+(-1-5)`

`H(x)=9x-6`

`G(x)=M(x)-N(x)`

\(G(x)=(-3x^5+9x^4+6x-1)-(3x^5-9x^4+3x-5)\)

`G(x)=-3x^5+9x^4+6x-1-3x^5+9x^4-3x+5`

`G(x)=(-3x^5-3x^5)+(9x^4+9x^4)+(6x-3x)+(-1+5)`

`G(x)=-6x^5+18x^4+3x+4`

`c,`

`H(x)=9x-6`

Hệ số cao nhất của đa thức: `9`

Hệ số tự do: `-6`

`G(x)=-6x^5+18x^4+3x+4`

Hệ số cao nhất của đa thức: `-6`

Hệ số tự do: `4`

`d,`

`H(-1)=9*(-1)-6=-9-6=-15`

`H(1)=9*1-6=9-6=3`

`G(1)=-6*1^5+18*1^4+3*1+4`

`G(1)=-6+18+3+4=12+3+4=15+4=19`

`G(0)=-6*0^5+18*0^4+3*0+4=4`

`H(-3/2)=9*(-3/2)-6=-27/2-6=-39/2`

`e,`

Đặt `H(x)=9x-6=0`

`-> 9x=0+6`

`-> 9x=6`

`-> x=6 \div 9`

`-> x=2/3`

Vậy, nghiệm của đa thức là `x=2/3.`