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a: Q(x)=3x^4+x^3+2x^2+x+1-2x^4+x^2-x+2
=x^4+x^2+3x^2+3
b: H(x)=2x^4-x^2+x-2-x^4+x^3-x^2+2
=x^4+x^3-2x^2+x
c: R(x)=2x^3+x^2+1+2x^4-x^2+x-2
=2x^4+2x^3+x-1
a: \(P\left(x\right)=x-2x^2+3x^5+x^4+x-1\)
\(=3x^5+x^4-2x^2+2x-1\)
\(Q\left(x\right)=3-2x-2x^2+x^4-3x^5-x^4+4x^2\)
\(=-3x^5+2x^2-2x+3\)
b: P(x)+Q(x)
\(=3x^5+x^4-2x^2+2x-1-3x^5+2x^2-2x+3\)
\(=x^4+2\)
P(x)-Q(x)
\(=3x^5+x^4-2x^2+2x-1+3x^5-2x^2+2x-3\)
\(=6x^5+x^4-4x^2+4x-4\)
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,R(x)=P(x)+Q(x)=-4x\(^4\)-2x+x\(^2\)+3x\(^3\)+1-2-3x\(^3\)+2x+x\(^5\)+5x\(^4\)
=x\(^5\)+(-4x\(^4\)+5x\(^4\))+(3x\(^3\)-3x\(^3\))+x\(^2\)+(-2x+2x)+(1-2)
=x\(^5\)+x\(^4\)+x\(^2\)-1
R(-1)=(-1)\(^5\)+(-1)\(^4\)+(-1)\(^2\)-1
=0
\(P\left(x\right)-Q\left(x\right)=\left(-2x+\frac{1}{2}x^2+3x^4-3x^2-3\right)-\left(3x^4+x^3-4x^2+1,5x^3-3x^4+2x+1\right)\\ P\left(x\right)-Q\left(x\right)=-2x+\frac{1}{2}x^2+3x^4-3x^2-3-3x^4-x^3+4x^2-1,5x^3+3x^4-2x-1\\ P\left(x\right)-Q\left(x\right)=\left(-2x-2x\right)+\left(\frac{1}{2}x^2-3x^2+4x^2\right)+\left(3x^4-3x^4+3x^4\right)+\left(-3-1\right)+\left(-x^3-1,5x^3\right)\\ P\left(x\right)-Q\left(x\right)=-4x+\frac{3}{2}x^2+3x^4-4-\frac{5}{2}x^3\)
\(R\left(x\right)+P\left(x\right)-Q\left(x\right)+x^2=2x^3-\frac{3}{2}x+1\\ \Rightarrow R\left(x\right)+\left(P\left(x\right)-Q\left(x\right)\right)+x^2=2x^3-\frac{3}{2}x+1\\ \Rightarrow R\left(x\right)-4x+\frac{3}{2}x^2+3x^4-4-\frac{5}{2}x^3+x^2=2x^3-\frac{3}{2}x+1\\ \Rightarrow R\left(x\right)-4x+\left(\frac{3}{2}x+x^2\right)+3x^4-4-\frac{5}{2}x^3=2x^3-\frac{3}{2}x+1\\ \Rightarrow R\left(x\right)-4x+\frac{5}{2}x^2+3x^4-4-\frac{5}{2}x^3=2x^3-\frac{3}{2}x+1\\ \Rightarrow R\left(x\right)=2x^3-\frac{3}{2}x+1+4x-\frac{5}{2}x^2-3x^4+4+\frac{5}{2}x^3\\ \Rightarrow R\left(x\right)=\left(2x^3+\frac{5}{2}x^3\right)+\left(\frac{-3}{2}x+4x\right)+\left(1+4\right)-\frac{5}{2}x^2-3x^4\\ \Rightarrow R\left(x\right)=\frac{9}{2}x^3+\frac{5}{2}x+5-\frac{5}{2}x^2-3x^4\)
\(P\left(x\right)+Q\left(x\right)=\left(2x^4+x^3-4x+5\right)+\left(x^4+3x^3+2x-1\right)\)
\(=2x^4+x^3-4x+5+x^4+3x^3+2x-1\)
\(=\left(2x^4+x^4\right)+\left(x^3+3x^3\right)+\left(-4x+2x\right)+\left(5-1\right)\)
\(=3x^4+4x^3-2x+4\)
\(R\left(x\right)+P\left(x\right)=x^4-2x^2+1\)
\(\Rightarrow R\left(x\right)=\left(x^4-2x^2+1\right)-P\left(x\right)\)
\(\Rightarrow R\left(x\right)=\left(x^4-2x^2+1\right)-\left(2x^4+x^3-4x+5\right)\)
\(\Rightarrow R\left(x\right)=x^4-2x^2+1-2x^4-x^3+4x-5\)
\(\Rightarrow R\left(x\right)=\left(x^4-2x^4\right)+\left(-2x^2\right)+\left(1-5\right)+\left(-x^3\right)+4x\)
\(\Rightarrow R\left(x\right)=-x^4-2x^2-4-x^3+4x\)