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If a polynomial function, f(x), with rational coefficients has roots 0, 4, and , what must also be a root of f(x)?.

Sagot :

The remaining root of f(x) will be 3 - [tex]\sqrt{11}[/tex]

To answer this question we have to find the conjugate of 3+[tex]\sqrt{11}[/tex],

For example, the conjugate of p+q will be p-q or vice versa.

Let us consider x= 2+[tex]\sqrt{11}[/tex]

To simplify the equation let us subtract 3 on both sides of the equation

x - 3 = 3+[tex]\sqrt{11}[/tex] - 3

x - 3 = [tex]\sqrt{11}[/tex]

Square both,

[tex](x - 3)^{2}[/tex] = 11

[tex](x - 3)^{2}[/tex] - 11 =0

Now we will use the principle of Square difference to factorize both sides-

[tex](d^{2} - e^{2}) = (d - e)(d + e)[/tex]

(x - 3) - 11 × (x - 3) + 11 = 0

on solving the equation we get,

(x - 3) - 1 = 0  let us consider this first equation

or (x - 3) + 11 = 0 let us consider this second equation

Add [tex]\sqrt{11}[/tex] on both sides of the first equation,

[tex]x - 3 = \sqrt{11}[/tex]

Subtract [tex]\sqrt{11}[/tex] on both sides of the second equation,

[tex]x - 3 = -\sqrt{11}[/tex]

Now by adding 3 on both sides of the first and second equation,

[tex]x = 3 + \sqrt{11}[/tex] or [tex]x = 3 - \sqrt{11}[/tex]

Therefore, the remaining root of f(x) will be = [tex]3 - \sqrt{11}[/tex]

To learn about Rational Coefficients,

https://brainly.com/question/19157484

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Complete Question - If a polynomial function f(x), with rational coefficients, has roots 0, 4, and 3 + [tex]\sqrt{11}[/tex], what must also be a root of f(x)?.

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