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Answer:
A polynomial function f of least degree = f = x³ - 7x² + 36
The function in standard form = f = x³ - 7x² + 36
Step-by-step explanation:
As given, Leading coefficient of polynomial function = 1
Also given, zeroes of the function = -2, 3, 6
for root -2 : (x - (-2)) = (x + 2)
for root 3 : ( x - 3)
for root 6 : (x - 6)
Therefore, the polynomial function becomes -
f = 1(x+2)(x-3)(x-6)
⇒f = (x+2)(x² - 6x - 3x + 18)
⇒f = (x+2)(x² - 9x + 18)
⇒f = x³ - 9x² + 18x + 2x² - 18x + 36
⇒f = x³ - 7x² + 36
The polynomial function in standard form is [tex]f(x) = x^3 - 8x^2 +9x +18[/tex]
The zeros of the polynomial are given as:
[tex]Zeroes = -2,3,6[/tex]
Rewrite as:
[tex]x= -2,3,6[/tex]
Express the zeroes as an equation
[tex]x= -2[/tex], [tex]x =3[/tex] and [tex]x =6[/tex]
Equate to 0
[tex]x + 2 = 0[/tex], [tex]x -3 = 0[/tex] and [tex]x -6 = 0[/tex]
Multiply the equations
[tex](x + 1) \times (x - 3) \times (x - 6) = 0 \times 0 \times 0[/tex]
[tex](x + 1) \times (x - 3) \times (x - 6) = 0[/tex]
Expand
[tex](x^2 - 3x + x - 3) \times (x - 6) = 0[/tex]
[tex](x^2 - 2x - 3) \times (x - 6) = 0[/tex]
Expand
[tex]x^3 - 2x^2 - 3x -6x^2 +12x +18 = 0[/tex]
Collect like terms
[tex]x^3 - 2x^2 -6x^2 - 3x +12x +18 = 0[/tex]
[tex]x^3 - 8x^2 +9x +18 = 0[/tex]
Express as a function
[tex]f(x) = x^3 - 8x^2 +9x +18[/tex]
Hence, the function in standard form is [tex]f(x) = x^3 - 8x^2 +9x +18[/tex]
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