Question

1.using identity, evaluate (43)³+(-18)³+(-25)³

2.find the remainder x⁴-x³+2x+1 is divided by 2x+1
3.find the value of k
x-1 is a factor of x³-2kx²+x+3​

Answer 1

Question 1

Let
`x=18` and
`y=25`. Then, we are required to compute:

`(x+y)^3 -x^3-y^3=x^3 +3x^2 y+3y^2 x+y^3-x^3-y^3=3xy(x+y)=\boxed{58050}`

Question 2

Using the remainder theorem, the remainder is equal to
`x^4 -x^3 +2x+1` computed at
`x=-1/2`. Doing so yields
`(-1/2)^4 -(-1/2)^3 +2(-1/2)+1=\boxed{3/16}`

Question 3

Using the factor theorem, the polynomial
`x^3 -2kx^2 +x+3` should equal zero when evaluated at
`x=1`.

`\implies 1-2k+1+3=0 \implies k=\boxed{5/2}`