a. \(P\left(x\right)=3x^5+5x-4x^4-2x^3+6+4x^2\)
\(=3x^5-4x^4-2x^3+4x^2+5x+6\)
* \(Q\left(x\right)=2x^4-x+3x^2-2x^3+\dfrac{1}{4}-x^5\)
\(=-x^5+2x^4-2x^3+3x^2-x+\dfrac{1}{4}\)
b. \(P\left(x\right)+Q\left(x\right)\)
\(=3x^5+5x-4x^4-2x^3+6+4x^2+2x^4-x+3x^2-2x^3+\dfrac{1}{4}-x^5\)
\(=\left(3x^5-x^5\right)+\left(-4x^4+2x^4\right)+\left(-2x^3-2x^3\right)+\left(4x^2+3x^2\right)+\left(5x-x\right)+\left(6+\dfrac{1}{4}\right)\)
\(=2x^5-2x^4-4x^3+7x^2+4x+\dfrac{25}{4}\)
* \(P\left(x\right)-Q\left(x\right)\)
\(=3x^5+5x-4x^4-2x^3+6+4x^2-2x^4+x-3x^2+2x^3-\dfrac{1}{4}+x^5\)
\(=\left(3x^5+x^5\right)+\left(-4x^4-2x^4\right)+\left(-2x^3+2x^3\right)+\left(4x^2-3x^2\right)+\left(5x+x\right)+\left(6-\dfrac{1}{4}\right)\)
\(=4x^5-6x^4+x^2+6x+\dfrac{23}{4}\)
c. Thay x = 1 lần lượt vào P(x), Q(x) :
\(P\left(-1\right)=3.\left(-1\right)^5+5.\left(-1\right)-4.\left(-1\right)^4-2.\left(-1\right)^3+6+4.\left(-1\right)^2=0\left(đpcm\right)\)
\(Q\left(-1\right)=2.\left(-1\right)^4-\left(-1\right)+3.\left(-1\right)^2-2.\left(-1\right)^3+\dfrac{1}{4}-\left(-1\right)^5=\dfrac{37}{4}\left(đpcm\right)\)
