Step-by-step explanation:
please read carefully. it says the polynomial of degree 4. that means the polynomial has the degree 4. and that means the highest exponent of the variable is 4.
further it says here that there are 2 roots (so, 2 solutions, where the functional value is 0) at x = 3. this is a double 0 solution.
then there is another 0 solution at x = 0, and a fourth zero solution at x = -4.
remember, a polynomial of the degree n must have n roots (n zero solutions).
we can construct this easily by building the whole function definition via multiplication terms (factors).
each factor represents one expression that turns 0 at the specified x value.
what expression turns 0 when x = 3 ?
well, (x - 3). there is no magic involved.
and we need that twice.
what expression turns 0 when x = 0 ?
well, simply x.
and what expression turns 0 when x = -4 ?
well, (x + 4).
so, the corresponding functional definition is then
P(x) = x(x-3)(x-3)(x+4)
that gives us a polynomial of degree 4 (highest exponent of x is 4) with the desired roots (0 function results at the specified x values).
but it has to go through the point (5, 18).
let's see :
P(5) = 5×(5-3)(5-3)(5+4) = 5×2×2×9 = 180.
aha ! too much !
how do we get the result down to just 18 ?
by dividing the whole thing by 10 !
dividing 0 by 10 is still 0, so this does not change our 0 solutions.
so, the final solution and correct polynomial is
P(x) = x(x-3)(x-3)(x+4)/10 = x(x-3)²(x+4)/10
oh, and if you want this in pure x terms without any factors, then we need to do the multiplications :
x×(x² - 6x + 9)(x+4)/10 =
(x³ - 6x² + 9x)(x+4)/10 =
(x⁴ - 6x³ + 9x² + 4x³ - 24x² + 36x)/10 =
(x⁴ - 2x³ - 15x² + 36x)/10
