Polynomials are expressions that uses variables, constants and exponents.
The polynomial is: [tex]\mathbf{P(x) = -9x^3 - 9x^2 + 9x + 2}[/tex]
The polynomial is given as:
[tex]\mathbf{P(x) = ax^3 + bx^2 + cx + d}[/tex]
[tex]\mathbf{y = 9x + 2}[/tex]
At (0,2), we have:
[tex]\mathbf{a(0)^3 + b(0)^2 + c(0) + d= 2}[/tex]
[tex]\mathbf{d= 2}[/tex]
At (-1,-7), we have:
[tex]\mathbf{a(-1)^3 + b(-1)^2 + c(-1) + d= -7}[/tex]
[tex]\mathbf{-a + b - c + d= -7}[/tex]
Substitute
[tex]\mathbf{-a + b - c + 2= -7}[/tex]
Subtract 2 from both sides
[tex]\mathbf{-a + b - c = -9}[/tex]
Recall that:
[tex]\mathbf{P(x) = ax^3 + bx^2 + cx + d}[/tex]
Integrate
[tex]\mathbf{P'(x) = 3ax^2 + 2bx + c}[/tex]
We have:
[tex]\mathbf{y = 9x + 2}[/tex]
Integrate
[tex]\mathbf{y' = 9}[/tex]
The above means that:
[tex]\mathbf{P'(0) = 9}[/tex]
So, we have:
[tex]\mathbf{P'(x) = 3ax^2 + 2bx + c}[/tex]
[tex]\mathbf{3a(0)^2 + 2b(0) + c = 9}[/tex]
[tex]\mathbf{c = 9}[/tex]
Substitute 9 for c in [tex]\mathbf{-a + b - c = -9}[/tex]
[tex]\mathbf{-a + b - 9 = -9}[/tex]
Add 9 to both sides
[tex]\mathbf{-a + b = 0}[/tex]
So, we have:
[tex]\mathbf{a = b }[/tex]
The line touches [tex]\mathbf{y = 9x + 2}[/tex] means that:
[tex]\mathbf{P'(-1) = 0}[/tex]
So, we have:
[tex]\mathbf{P'(x) = 3ax^2 + 2bx + c}[/tex]
[tex]\mathbf{3a(-1)^2 - 2b + c = 0}[/tex]
[tex]\mathbf{3a - 2b + c = 0}[/tex]
Substitute 9 for c
[tex]\mathbf{3a - 2b + 9 = 0}[/tex]
Recall that: [tex]\mathbf{a = b }[/tex]
[tex]\mathbf{3a - 2a + 9 = 0}[/tex]
[tex]\mathbf{a + 9 = 0}[/tex]
[tex]\mathbf{a =- 9}[/tex]
So, we have:
[tex]\mathbf{a =b = - 9}[/tex]
Substitute [tex]\mathbf{a =b = - 9}[/tex], [tex]\mathbf{c = 9}[/tex] and [tex]\mathbf{d= 2}[/tex] in [tex]\mathbf{P(x) = ax^3 + bx^2 + cx + d}[/tex]
So, we have:
[tex]\mathbf{P(x) = -9x^3 - 9x^2 + 9x + 2}[/tex]
Hence, the polynomial is:
[tex]\mathbf{P(x) = -9x^3 - 9x^2 + 9x + 2}[/tex]
Read more about polynomials at:
https://brainly.com/question/11536910
