`@` `\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(x) = -2x^2 + 4x^4 – 9x^3 + 3x^2 – 5x + 3

=4x^4-9x^3+x^2-5x+3

Q(x) = 5x^4 – x^3 + x^2 – 2x^3 + 3x^2 – 2 – 5x

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

b)

P(x)

-bậc:4

-hệ số tự do:3

-hệ số cao nhất:4

Q(x)

-bậc :4

-hệ số tự do :-2

-hệ số cao nhất:5

`P(x)=`\( 2x^4 + 3x^3 + 3x^2 - x^4 - 4x + 2 - 2x^2 + 6x\)

`= (2x^4-x^4)+3x^3+(3x^2-2x^2)+(-4x+6x)+2`

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

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

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

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

`P(x)+Q(x)=(x^4+3x^3+x^2+2x+2)+(x^4+x^3+2x^2+2x+1)`

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

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

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

`@`\(\text{dn inactive.}\)

P(x)=x^4+3x^3+x^2+2x+2

Q(x)=x^4+x^3+2x^2+2x+1

P(x)+Q(x)=2x^4+4x^3+3x^2+4x+3

P(x) = 2x⁴ + 3x³ + 3x² - x⁴ - 4x + 2 - 2x² + 6x

Q(x) = x⁴ + 3x² + 5x - 1 - x² - 3x + 2 + x³

P(x)+Q(x) = 2x⁴ + 3x³ + 3x² - x⁴ - 4x + 2 - 2x² + 6x + x⁴ + 3x² + 5x - 1 - x² - 3x + 2 + x³

P(x)+Q(x) = (2x⁴-x⁴+x⁴) + (3x³+x³) + (3x²-2x²+3x²-x²) - (4x-6x-5x+3x) +(2-1+2)

P(x)+Q(x) = 4x³+3x²-4x+3

Câu 1: 

a) Ta có: \(P\left(x\right)=x^5-3x^2+7x^4-9x^3+x^2-\dfrac{1}{4}x\)

\(=x^5+7x^4-9x^3-2x^2-\dfrac{1}{4}x\)

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

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

Câu 1:

b) Ta có: P(x)+Q(x)

\(=x^5+7x^4-9x^3-2x^2-\dfrac{1}{4}x-x^5+5x^4-2x^3+4x^2-\dfrac{1}{4}\)

\(=12x^4-11x^3+2x^2-\dfrac{1}{4}x-\dfrac{1}{4}\)

\(Q-P=3x^3-3x^2+8x-5-5x^2+3x-2\)

\(Q-P=3x^3+\left(-3x^2-5x^2\right)+\left(8x+3x\right)+\left(-5-2\right)\)

\(Q-P=3x^3-8x^2+11x-7\)