Polynomials are expressions that uses variables, constants and exponents.
The correct statement is: (c) The graph of the function is positive on (-∞, -7)
The given parameters are:
[tex]\mathbf{(Root, Multiplicity) = (-7,2)}[/tex]
This means that: [tex]\mathbf{(x + 7)^2}[/tex]
[tex]\mathbf{(Root, Multiplicity) = (-1,1)}[/tex]
This means that: [tex]\mathbf{(x + 1)^1}[/tex]
[tex]\mathbf{(Root, Multiplicity) = (2,4)}[/tex]
This means that: [tex]\mathbf{(x - 1)^4}[/tex]
[tex]\mathbf{(Root, Multiplicity) = (4,1)}[/tex]
This means that: [tex]\mathbf{(x - 4)^1}[/tex]
So, the polynomial is:
[tex]\mathbf{P(x) = a(x + 7)^4(x + 1)^1(x - 1)^4(x - 4)^1}[/tex]
[tex]\mathbf{P(x) = a(x + 7)^4(x + 1)(x - 1)^4(x - 4)}[/tex]
Rearrange
[tex]\mathbf{P(x) = a(x + 7)^4(x - 1)^4(x + 1)(x - 4)}[/tex]
Next, we test the options
(a) The graph of the function is positive on (2,4)
Use x = 3
[tex]\mathbf{P(3) = a(3 + 7)^4(3 - 1)^4(3 + 1)(3 - 4)}[/tex]
[tex]\mathbf{P(3) = -640000a}[/tex]
The above is a negative number
So, (a) is false
(b) The graph of the function is negative on (4, ∞)
Use x = 6
[tex]\mathbf{P(6) = a(6 + 7)^4(6 - 1)^4(6 + 1)(6 - 4)}[/tex]
[tex]\mathbf{P(6) = 249908750a}[/tex]
The above is a positive number.
So, (b) is false
(c) The graph of the function is positive on (-∞, -7)
Use x = -8
[tex]\mathbf{P(-8) = a(-8 + 7)^4(-8 - 1)^4(-8 + 1)(-8 - 4)}[/tex]
[tex]\mathbf{P(-8) = 551124a}[/tex]
The above is a positive number.
So, (c) is false
Hence, the correct statement is: (c) The graph of the function is positive on (-∞, -7)
Read more about polynomials at:
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