Bài 1:
a) \(A\left(x\right)+B\left(x\right)=\left(-x^3+x^2-5x+1\right)+\left(x^3+4x-5\right)\)
\(=-x^3+x^2-5x+1+x^3+4x-5\)
\(=\left(-x^3+x^3\right)+x^2+\left(-5x+4x\right)+\left(1-5\right)\)
\(=x^2-x-4\)
b) \(A\left(x\right)-B\left(x\right)=\left(-x^3+x^2-5x+1\right)-\left(x^3+4x-5\right)\)
\(=-x^3+x^2-5x+1-x^3-4x+5\)
\(=\left(-x^3-x^3\right)+x^2+\left(-5x-4x\right)+\left(1+5\right)\)
\(=-2x^3+x^2-9x+6\)
Bài 2
* \(P+Q=\left(x^5+7x^3+1\right)+\left(x^3-4x^5+2\right)\)
\(=x^5+7x^3+1+x^3-4x^5+2\)
\(=\left(x^5-4x^5\right)+\left(7x^3+x^3\right)+\left(1+2\right)\)
\(=-3x^5+8x^3+3\)
* \(P-Q=\left(x^5+7x^3+1\right)-\left(x^3-4x^5+2\right)\)
\(=x^5+7x^3+1-x^3+4x^5-2\)
\(=\left(x^5+4x^5\right)+\left(7x^3-x^3\right)+\left(1-2\right)\)
\(=5x^5+6x^3-1\)
Bài 3
a) \(A=3x^2+x+x^4-x^3-x^2+2x\)
\(=x^4-x^3+\left(3x^2-x^2\right)+\left(x+2x\right)\)
\(=x^4-x^3+2x^2+3x\)
\(B=x^4+2x^2+x^3\)
\(=x^4+x^3+2x^2\)
b) * Đa thức A có:
- Bậc: 4
- Hệ số tự do: 0
- Hệ số cao nhất: 1
* Đa thức B có:
- Bậc: 4
- Hệ số tự do: 0
- Hệ số cao nhất: 1
c) \(A+B=\left(x^4-x^3+2x^2+3x\right)+\left(x^4+x^3+2x^2\right)\)
\(=x^4-x^3+2x^2+3x+x^4+x^3+2x^2\)
\(=\left(x^4+x^4\right)+\left(-x^3+x^3\right)+\left(2x^2+2x^2\right)+3x\)
\(=2x^4+4x^2+3x\)
\(A-B=\left(x^4-x^3+2x^2+3x\right)-\left(x^4+x^3+2x^2\right)\)
\(=x^4-x^3+2x^2+3x-x^4-x^3-2x^2\)
\(=\left(x^4-x^4\right)+\left(-x^3-x^3\right)+\left(2x^2-2x^2\right)+3x\)
\(=-2x^3+3x\)
\(B-A=\left(x^4+x^3+2x^2\right)-\left(x^4-x^3+2x^2+3x\right)\)
\(=x^4+x^3+2x^2-x^4+x^3-2x^2-3x\)
\(=\left(x^4-x^4\right)+\left(x^3+x^3\right)+\left(2x^2-2x^2\right)-3x\)
\(=2x^3-3x\)
Bài 4:
a) \(P=3x-2x^2-2x+6x^3\)
\(=\left(3x-2x\right)-2x^2+6x^3\)
\(=x-2x^2+6x^3\)
\(Q=x^2-x-2x^3+4\)
\(=4-x+x^2-2x^3\)
b) \(R=P+Q\)
\(=\left(x-2x^2+6x^3\right)+\left(4-x+x^2-2x^3\right)\)
\(=x-2x^2+6x^3+4-x+x^2-2x^3\)
\(=\left(6x^3-2x^3\right)+\left(-2x^2+x^2\right)+\left(x-x\right)+4\)
\(=4x^3-x^2+4\)
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\(H=P-Q\)
\(=\left(x-2x^2+6x^3\right)-\left(4-x+x^2-2x^3\right)\)
\(=x-2x^2+6x^3-4+x-x^2+2x^3\)
\(\left(6x^3+2x^3\right)+\left(-2x^2-x^2\right)+\left(x+x\right)-4\)
\(=8x^3-3x^2+2x-4\)
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e) \(R\left(4\right)=4.4^3-4^2+4\)
\(=256-16+4\)
\(=244\)
\(H\left(-5\right)=8.\left(-5\right)^3-3.\left(-5\right)^2+2.\left(-5\right)-4\)
\(=-1000-75-10-4\)
\(=-1089\)
5:
a: A(x)=5x^4-2x^3+x^2-4
B(x)=-x^5+7x^4-9x^3-3x^2+7
b: =>-C(x)=-B(x)-A(x)
=>C(x)=A(x)+B(x)
=5x^4-2x^3+x^2-4-x^5+7x^4-9x^3-3x^2+7
=-x^5+12x^4-11x^3-2x^2+3
c: Hệ số cao nhất là -1
Hệ số lũy thừa bậc 2 là -2
Hệ số tự do là 3
Bậc là 5
Bài 5
a) \(A\left(x\right)=5x^4-2x^3+x^2-4\)
\(B\left(x\right)=-x^5+7x^4-3x^2-9x^3+7\)
\(=-x^5+7x^4-9x^3-3x^2+7\)
b) \(2C\left(x\right)+A\left(x\right)=3C\left(x\right)-B\left(x\right)\)
\(\Rightarrow3C\left(x\right)-2C\left(x\right)=A\left(x\right)+B\left(x\right)\)
\(\Rightarrow C\left(x\right)=A\left(x\right)+B\left(x\right)\)
\(=\left(5x^4-2x^3+x^2-4\right)+\left(-x^5+7x^4-9x^3-3x^2+7\right)\)
\(=5x^4-2x^3+x^2-4-x^5+7x^4-9x^3-3x^2+7\)
\(=-x^5+\left(5x^4+7x^4\right)+\left(-2x^3-9x^3\right)+\left(x^2-3x^2\right)+\left(-4+7\right)\)
\(=-x^5+12x^4-11x^3-2x^2+3\)
c) C(x) có:
- Hệ số cao nhất: -1
- Hệ số lũy thừa bậc 2: -2
- Hệ số tự do: 3
- Bậc: 5
