Cho 2 đa thức :
f(x) +g(x) = 2x^4 + 5x^2 - 3x
f(x) - g(x) = x^4 - x^2 +2x
Tìm f(x) và g(x)
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![](https://rs.olm.vn/images/avt/0.png?1311)
a)f(x)+g(x)=\(x^5-4x^4-2x^2-7-2x^5+6x^4-2x^2+6.\)
=\(-x^5+2x^4-4x^2-1\)
f(x)-g(x)=\(x^5-4x^4-2x^2-7+2x^5-6x^4+2x^2-6\)
=\(3x^5-10x^4-13\)
b)f(x)+g(x)=\(5x^4+7x^3-6x^2+3x-7-4x^4+2x^3-5x^2+4x+5\)
=\(x^4+9x^3-11x^2+7x-2\)
f(x)-g(x)=\(5x^4+7x^3-6x^2+3x-7+4x^4-2x^3+5x^2-4x-5\)
=\(9x^4+5x^3-x^2-x-12\)
a )
\(f\left(x\right)+g\left(x\right)=x^5-4x^4-2x^2-7+-2x^5+6x^4-2x^2+6\)
\(\Rightarrow f\left(x\right)+g\left(x\right)=\left(x^5-2x^5\right)+\left(6x^4-4x^4\right)-\left(2x^2+2x^2\right)+\left(6-7\right)\)
\(\Rightarrow f\left(x\right)+g\left(x\right)=-x^5+2x^4-4x^2-1\)
\(f\left(x\right)-g\left(x\right)=x^5-4x^4-2x^2-7-\left(-2x^5+6x^4-2x^2+6\right)\)
\(\Rightarrow f\left(x\right)-g\left(x\right)=x^5-4x^4-2x^2-7+2x^5-6x^4+2x^2-6\)
\(\Rightarrow f\left(x\right)-g\left(x\right)=\left(x^5+2x^5\right)-\left(4x^4+6x^4\right)+\left(2x^2-2x^2\right)-\left(6+7\right)\)
\(\Rightarrow f\left(x\right)-g\left(x\right)=3x^5-10x^4-13\)
![](https://rs.olm.vn/images/avt/0.png?1311)
f(x) + g(x) = 2x4 + 2x2
f(x) - g(x) = x4 - x2 + 2x
suy ra : f(x) = [ ( 2x4 + 2x2 ) + ( x4 - x2 + 2x ) ] : 2 = \(\frac{3x^4+x^2+2x}{2}\)
g(x) = [ ( 2x4 + 2x2 ) - ( x4 - x2 + 2x ) ] : 2 = \(\frac{x^4+3x^2-2x}{2}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có:\(f\left(x\right)-h\left(x\right)=g\left(x\right)\Leftrightarrow h\left(x\right)=f\left(x\right)-g\left(x\right)\)
\(\Leftrightarrow h\left(x\right)=\left(2x^4+5x^3-x+8\right)-\left(x^4-x^2+3x+9\right)\)
\(=2x^4+5x^3-x+8-x^4-x^2-3x-9\)
\(=x^4+5x^3+x^2-4x-1.\)
Vậy, đa thức cần tìm là: \(h\left(x\right)=x^4+5x^3+x^2-4x-1.\)
Ta có: \(h\left(x\right)-g\left(x\right)=f\left(x\right)\Leftrightarrow h\left(x\right)=f\left(x\right)+g\left(x\right)\)
\(\Leftrightarrow h\left(x\right)=\left(2x^4+5x^3-x+8\right)+\left(x^4-x^2+3x+9\right)\)
\(=2x^4+5x^3-x+8+x^4-x^2+3x+9\)
\(=3x^4+5x^3-x^2+2x+17\)
Vậy, đa thức cần tìm là:\(h\left(x\right)=3x^4+5x^3-x^2+2x+17.\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(f\left(x\right)=x^5-4x^4-2x^2-7\)
\(g\left(x\right)=-2x^5+6x^4-2x^2+6\)
\(f\left(x\right)+g\left(x\right)=-x^5+2x^4-4x^2-1\)
\(f\left(x\right)-g\left(x\right)=3x^5-10x^4-13\)
Xét [\(f\left(x\right)+g\left(x\right)\)]+[\(f\left(x\right)-g\left(x\right)\)]=\(\left[2x^4+5x^2-3x\right]\)+\(\left[x^4-x^2+2x\right]\)
\(2f\left(x\right)=2x^4+5x^2-3x+x^4-x^2+2x\)
\(2f\left(x\right)=3x^4+4x^2-x\)
\(\Rightarrow f\left(x\right)=\dfrac{3x^4+4x^2-x}{2}\)
\(\Rightarrow f\left(x\right)=\dfrac{3}{2}x^4+2x^2-\dfrac{1}{2}x\)
Xét \(\left[f\left(x\right)+g\left(x\right)\right]-\left[f\left(x\right)-g\left(x\right)\right]=\)\(\left[2x^4+5x^2-3x\right]\)\(-\)\(\left[x^4-x^2+2x\right]\)
\(2g\left(x\right)=\)\(2x^4+5x^2-3x-x^4+x^2-2x\)
\(2g\left(x\right)=x^4+6x^2-5x\)
\(\Rightarrow g\left(x\right)=\dfrac{x^4+6x^2-5x}{2}\)
\(\Rightarrow g\left(x\right)=\dfrac{1}{2}x^4+3x^2-\dfrac{5}{2}x\)