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Suppose p(x) is a polynomial of smallest possible degree such that: bullet p(x) has rational coefficients bullet p(−3)=p(sqrt7)=p(1−sqrt6)=0 bullet p(−1)=8 determine the value of p(0).

Answer :

ukshedrack

Answer:

p(0) = 28.07

Step-by-step explanation:

Since p(-3)=0, it means x= -3 is a root of the polynomial. And the factor is x-(-3) = x + 3

Also, p(sqrt7)=0, meaning that x= [tex]\sqrt{7}[/tex] is a root of the polynomial. And the factor is x - [tex]\sqrt{7}[/tex]

Lastly, p(1-sqrt6)=0, meaning that x= [tex]1-\sqrt{6}[/tex] is a root of the polynomial. And the factor is x - [tex](1-\sqrt{6})[/tex]

The polynomial p(x) = K(x+3)(x-√7)(x-(1-√6))

But p(-1) = 8. That is when x = -1, p(x) = 8. So we use this to find K

[tex]8 = K(-1 + 3)(-1 -\sqrt{7})(-1-(1-\sqrt{6})\\ \\8 = K(2)(-1 -\sqrt{7})(-2 +\sqrt{6}) = K(2)(-3.6457)(0.4495)\\\\8 = -3.2775K\\\\\frac{8}{-3.2775} = K\\\\K = -2.4409[/tex]

K ≅ -2.44

[tex]p(x) = -2.44(x + 3)(x -\sqrt{7})(x-(1-\sqrt{6})\\\\p(0) = -2.44(0+3)(0-\sqrt{7})(0-(1-\sqrt{6})\\\\p(0) = -2.44(3)(-2.6457)(1.4495) = 28.0716\\[/tex]

p(0) ≅ 28.07

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