Answer:
f(x) = 2x³ - 12x² - 2x + 60
Step-by-step explanation:
Given that the zeros are x = - 2, x = 3, x = 5 , then the factors are
(x + 2), (x - 3), (x - 5)
The polynomial is then the product of the factors, that is
f(x) = a(x + 2)(x - 3)(x - 5) ← a is a multiplier
To find a substitute (6, 48) into the polynomial
48 = a(8)(3)(1) = 24a ( divide both sides by 24 )
a = 2 , thus
f(x) = 2(x + 2)(x - 3)(x - 5) ← expand the last 2 factors using FOIL
= 2(x + 2)(x² - 8x + 15) ← distribute the parenthesis
= 2(x³ - 8x² + 15x + 2x² - 16x + 30)
= 2(x³ - 6x² - x + 30) ← distribute by 2
= 2x³ - 12x² - 2x + 60
Using the factor theorem, it is found that the polynomial function is given by:
[tex]p(x) = 2(x^3 - 6x^2 - x + 30)[/tex]
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Factor theorem:
Given a polynomial f(x), this polynomial has roots [tex]x_{1}, x_{2}, x_{n}[/tex] such that it can be written as: [tex]a(x - x_{1})*(x - x_{2})*...*(x-x_n)[/tex], in which a is the leading coefficient.
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The polynomial is given by:
[tex]p(x) = a(x - x_1)(x - x_2)(x-x_3)[/tex]
[tex]p(x) = a(x - (-2))(x - 3)(x - 5)[/tex]
[tex]p(x) = a(x + 2)(x - 3)(x - 5)[/tex]
[tex]p(x) = a(x^2 - x - 6)(x - 5)[/tex]
[tex]p(x) = a(x^3 - 6x^2 - x + 30)[/tex]
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[tex]p(x) = a(x^3 - 6x^2 - x + 30)[/tex]
[tex]48 = a(6^3 - 6(6)^2 - (6) + 30)[/tex]
[tex]24a = 48[/tex]
[tex]a = \frac{48}{24}[/tex]
[tex]a = 2[/tex]
Thus, the polynomial is given by:
[tex]p(x) = 2(x^3 - 6x^2 - x + 30)[/tex]
A similar question is given at https://brainly.com/question/24380382
