Câu1. Cho các đa thức:
????(????)=????^5−3????^2+7????^4−9????^3+????^2-1/4????;
????(????)=5????^4−????^5+????^2−2????^3+3????^2−1/4
a) Thu gọn và sắp xếp các hạng tử của mỗi đa thức theo lũy thừa giảm dần của biến.
b) Tính P(x) + Q(x) và P(x) – Q(x)
c) Tính P(1), Q(0)
Giải thích các bước giải:
`a)`
`P(x)=x^5-3x^2+7x^4-9x^3+x^2-(1)/(4)x`
`->P(x)=P(x)=x^5+7x^4-9x^3-3x^2+x^2-(1)/(4)x`
`->P(x)=x^5+7x^4-9x^3-2x^2-(1)/(4)x`
`Q(x)=5x^4-x^5+x^2-2x^3+3x^2-(1)/(4)`
`->Q(x)=-x^5+5x^4-2x^3+4x^2-(1)/(4)`
`b)`
`P(x)+Q(x)=x^5+7x^4-9x^3-2x^2-(1)/(4)x-x^5+5x^4-2x^3+4x^2-(1)/(4)`
`->P(x)+Q(x)=12x^4-11x^3+2x^2-(1)/(4)x-(1)/(4)`
`P(x)-Q(x)=x^5+7x^4-9x^3-2x^2-(1)/(4)x-(-x^5+5x^4-2x^3+4x^2-(1)/(4))`
`->P(x)-Q(x)=x^5+7x^4-9x^3-2x^2-(1)/(4)x+x^5-5x^4+2x^3-4x^2+(1)/(4)`
`->P(x)-Q(x)=2x^5+2x^4-7x^3-6x^2-(1)/(4)x+(1)/(4)`
`c)`
`P(1)=1^5+7.1^4-9.1^3-2.1^2-(1)/(4).1=1+7-9-2-(1)/(4)=(-13)/(4)`
`Q(0)=-0^5+5.0^4-2.0^3+4.0^2-(1)/(4)=0+0+0+0-(1)/(4)=(-1)/(4)`
`a)`
`P(x)`
`=x^5 – 3x^2 + 7x^4 – 9x^3 + x^2 – (1)/(4)x`
`=x^5 + 7x^4 – 9x^3 + (x^2 – 3x^2) – (1)/(4)x`
`=x^5 + 7x^4 – 9x^3 – 2x^2 – (1)/(4)x`
`Q(x)`
`=5x^4 – x^5 + x^2 – 2x^3 + 3x^2 – (1)/(4)`
`=-x^5 + 5x^4 – 2x^3 + (x^2 + 3x^2) – (1)/(4)`
`=-x^5 + 5x^4 – 2x^3 + 4x^2 – (1)/(4)`
`b)`
`P(x) + Q(x)`
`=(x^5 + 7x^4 − 9x^3 – 2x^2 – (1)/(4)x)+(−x^5 + 5x^4 −2x^3 + 4x^2 – (1)/(4))`
`=x^5 + 7x^4 − 9x^3 – 2x^2 – (1)/(4)x − x^5 + 5x^4 – 2x^3 + 4x^2 – (1)/(4)`
`=(x^5 – x^5) + (7x^4 + 5x^4) + (-9x^3 −2x^3) + (4x^2 – 2x^2) – (1)/(4)x – (1)/(4)`
`= 12x^4 – 11x^3 + 2x^2 – (1)/(4)x – (1)/(4)`
`P(x) – Q(x)`
`=(x^5 + 7x^4 – 9x^3 – 2x^2 – (1)/(4) x)-(-x^5 + 5x^4 – 2x^3 + 4x^2 – (1)/(4))`
`=x^5 + 7x^4 – 9x^3 – 2x^2 – (1)/(4) x + x^5 – 5x^4 + 2x^3 – 4x^2 + (1)/(4)`
`=(x^5 + x^5)+(7x^4 – 5x^4)+(2x^3 −9x^3)+(-2x^2 – 4x^2) – (1)/(4)x + (1)/(4)`
`=2x^5 + 2x^4 – 7x^3 – 6x^2 – (1)/(4) x + (1)/(4)`
`c)`
`P(1)`
`= 1^5 + 7.1^4 – 9.1^3 – 2.1^2 – (1)/(4).1`
`= 1 + 7.1 – 9.1 – 2.1 – (1)/(4)`
`= 1 + 7 – 9 – 2 – (1)/(4)`
`= -3 – (1)/(4)`
`= -(12+1)/4`
`= (-13)/(4)`
`Q(0)`
`= -0^5 + 5.0^4 – 2.0^3 + 4.0^2 – (1)/(4)`
`= 0 + 5.0 – 2.0 + 4.0 – (1)/(4)`
`= 0+0-0+0-(1)/(4)`
`= (-1)/(4)`