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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: \(A\left(x\right)=-5x^4+2x^2-8x+\dfrac{1}{2}+4x^4+2x^3-5x^2-6x+\dfrac{3}{2}=-x^4+2x^3-3x^2-14x+2\)
\(B\left(x\right)=-5x^4+2x^2-8x+\dfrac{1}{2}-4x^4-2x^3+5x^2+6x-\dfrac{3}{2}=-9x^4-2x^3+7x^2-2x-1\)
\(P\left(x\right)+Q\left(x\right)=\left(2x^4+x^3-4x+5\right)+\left(x^4+3x^3+2x-1\right)\)
\(=2x^4+x^3-4x+5+x^4+3x^3+2x-1\)
\(=\left(2x^4+x^4\right)+\left(x^3+3x^3\right)+\left(-4x+2x\right)+\left(5-1\right)\)
\(=3x^4+4x^3-2x+4\)
\(R\left(x\right)+P\left(x\right)=x^4-2x^2+1\)
\(\Rightarrow R\left(x\right)=\left(x^4-2x^2+1\right)-P\left(x\right)\)
\(\Rightarrow R\left(x\right)=\left(x^4-2x^2+1\right)-\left(2x^4+x^3-4x+5\right)\)
\(\Rightarrow R\left(x\right)=x^4-2x^2+1-2x^4-x^3+4x-5\)
\(\Rightarrow R\left(x\right)=\left(x^4-2x^4\right)+\left(-2x^2\right)+\left(1-5\right)+\left(-x^3\right)+4x\)
\(\Rightarrow R\left(x\right)=-x^4-2x^2-4-x^3+4x\)
`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
`@` `\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`
Ta có: \(P\left(x\right)=-2x^4-7x+\frac{1}{2}-3x^4+2x^2-x\)
\(=-5x^4+2x^2-8x+\frac{1}{2}\)
Ta có: \(Q\left(x\right)=3x^3+4x^4-5x^2-x^3-6x+\frac{3}{2}\)
\(=4x^4+2x^3-5x^2-6x+\frac{3}{2}\)
Ta có: R(x)=P(x)-Q(x)
\(=-5x^4+2x^2-8x+\frac{1}{2}-4x^4-2x^3+5x^2+6x-\frac{3}{2}\)
\(=-9x^4-2x^3+7x^2-2x-1\)
Thay x=-1 vào đa thức \(R\left(x\right)=-9x^4-2x^3+7x^2-2x-1\), ta được:
\(R\left(-1\right)=-9\cdot\left(-1\right)^4-2\cdot\left(-1\right)^3+7\cdot\left(-1\right)^2-2\cdot\left(-1\right)-1\)
\(=-9\cdot1+2+7+2-1\)
\(=-9+10=1\)
Vậy: x=-1 không là nghiệm của đa thức R(x)=P(x)-Q(x)
a: P(x)=x^4-2x^4-5x^3-7x^2+2x-1
=-x^4-5x^3-7x^2+2x-1
Q(x)=3x^4-2x^4+5x^3+6x^2-2x+5
=x^4+5x^3+6x^2-2x+5
P(\(x\)) = \(x^4\) - 2\(x^3\) - 3\(x^2\) + 7\(x\) - 2
Q(\(x\)) = \(x^4\) + \(x^3\) - 2\(x\) + 1
P(\(x\)) + Q(\(x\)) = \(x^4\) - 2\(x^3\) - 3\(x^2\) + 7\(x\)- 2 + \(x^4\) + \(x^3\) - 2\(x\) + 7\(x\) - 2
P(\(x\)) + Q(\(x\)) = ( \(x^4\) + \(x^4\)) - (2\(x^3\) - \(x^3\)) - 3\(x^2\) + ( 7\(x\) - 2\(x\)) - (2-1)
P(\(x\)) +Q(\(x\)) =2 \(x^4\) - \(x^3\) - 3\(x^2\)+ 5\(x\) - 1
P(\(x\)) - Q(\(x\)) = \(x^4\) -2 \(x^3\)-3\(x^2\) +7\(x\) - 2 - \(x^4\) - \(x^3\) +2\(x\) - 1
P(\(x\)) -Q(\(x\)) = (\(x^4\) - \(x^4\)) - (2\(x^3\) + \(x^3\)) - 3\(x^2\) + ( \(7x+2x\)) - ( 2 + 1)
P(\(x\)) -Q(\(x\)) = - 3\(x^3\) - 3\(x^2\)+ 9\(x\) - 3