It's a polynomial. You don't know if it has any term in x², x or any constants.

Since term containing the highest power of x is 8x³,

Let the polynomial P(x) = 8x³ + Ax² + Bx + C
We use the info to solve for the coefficients to determine what P(x) actually is.

Since two of the roots of P(x) = 0 are 0 and 2, then substitute x = 0 and x = 2 separately.

P(0) = 0
8(0³) + A(0²) + B(0) + C = 0
C = 0

So there is no constant term.

Next,
P(2) = 0
And we know C = 0 already.

So,
8(2³) + A(2²) + B(2) + 0 = 0
64 + 4A + 2B = 0 ①


Next, it is given that the remainder of 35 when divided by (2x + 1)
For (2x + 1), the value of x to use is x = -½ (such that this term becomes 0 when we apply the remainder theorem)

By the remainder theorem,

P(-½) = 35
8(-½)³ + A(-½)² + B(-½) + 0 = 35
-1 + ¼A - ½B + 0 = 35
¼A - ½B = 36
Multiply both sides by 4,
A - 2B = 144 ②


Then, do ① + ②

64 + 4A + 2B + A - 2B = 0 + 144
5A + 64 = 144
5A = 80
A = 16

Then, sub A = 16 back into ②

16 - 2B = 144
-2B = 144 - 16 = 128
B = 128/-2 = -64

So,

P(x) = 8x³ + 16x² - 64x

thx :) i understand it now

Normally I teach them the other method. Essentially the exact reverse of quadratic or polynomial equation solving.

Because x = 0 and x = 2 are roots of the equation f(x) = 0, then (x - 0) and (x - 2) are factors of f(x); since the term containing the highest power of f(x) is 8x3, then we need another linear term for this of the form (ax + b), but clearly a = 8.

x (x - 2) (8x + b) = 0

Of course this is zero because we are solving for f(x) = 0, but the original equation for f(x) did not equate to zero (it was zero because we equated them to zero).

f(x) = x (x - 2) (8x + b)

Considering the remainder 35 when divided by (2x + 1),

f(-1/2) = 35
-1/2 (-5/2) (-4 + b) = 35
5/4 (-4 + b) = 35
-4 + b = 28
b = 32

Then,

f(x) = x (x - 2) (8x + 32)
f(x) = x (8x2 + 32x - 16x - 64)
f(x) = 8x3 + 16x2 - 64x

(I just realised that it should be P(x) and not f(x))

Letting P(x) = f(x),

P(x) = 8x3 + 16x2 - 64x

Yup, it works for this question.

However, if there were only 1 real root and 2 complex roots, it might not work so smoothly though.

Example question :

Term containing the highest power of x is 8x³, and 2 is a solution/root of the equation P(x) = 0. P(x) gives a remainder of 76 when divided by (x - 3) and a remainder of 26¾ when divided by (2x - 5)

Find the expression/polynomial P(x)