Answer:
The polynomial in factored form is:
P(x) = (x-10)(x+1)(x+1)
or, we can write it in expanded form as:
P(x) = x^3 - 8x^2 - 11x + 40.
Explanation:
If the x-intercepts are at (10,0) and (-1,0), then the polynomial must have factors of (x-10) and (x+1). If the y-intercept is at (0,-40), then the constant term must be -40.
To satisfy the given requirements, we can write the polynomial as:
P(x) = (x-10)(x+1)(x-a)
where a is a constant that we need to determine. To find a, we can use the fact that the leading coefficient is 1.
Expanding the expression above, we get:
P(x) = (x^2 - 9x - 10)(x-a)
= x^3 - ax^2 - 9x^2 + 9ax - 10x + 10a
= x^3 - (a+9)x^2 + (9a-10)x + 10a
Since the leading coefficient is 1, we know that the coefficient of x^3 is 1. Therefore, we must have:
1 = 1*(-a-9)*(9a-10)
Simplifying and solving for a, we get:
a = -1
Therefore, the polynomial in factored form is:
P(x) = (x-10)(x+1)(x+1)
or, we can write it in expanded form as:
P(x) = x^3 - 8x^2 - 11x + 40.