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Now, suppose one of the roots of the polynomial function is complex. The roots of the function are 2 + i, and 5. Write the equation for this polynomial function.


Which of the following must also be a root of the function?


–3


–5


2 – i


2i

Verified Answer :

InesWalston

Answer-

The equation of polynomial- \(x^3-9x^2+25x-25=0\)

\((2-i)\) must also be a root of the function.

Solution-

Complex Conjugate Root Theorem-

For a real coefficient polynomial P(x), if \((a+bi)\) is a root then \((a-bi)\) will also be a root of P(x).

As \(2+i\) is a root of the function, so \(2-i\) will also a root of the function.

All the roots of the functions are,

\(5,\ 2+i,\ 2-i\)

So,

\(\Rightarrow P(x)=(x-5)(x-(2+i))(x-(2-i))=0\)

\(\Rightarrow (x-5)(x-2-i)(x-2+i)=0\)

\(\Rightarrow (x-5)((x-2)+(i))((x-2)-(i))=0\)

\(\Rightarrow (x-5)((x-2)^2-(i)^2)=0\)

\(\Rightarrow (x-5)(x^2-4x+4+1)=0\)

\(\Rightarrow (x-5)(x^2-4x+5)=0\)

\(\Rightarrow xx^2+x\left(-4x\right)+x\cdot \:5+\left(-5\right)x^2+\left(-5\right)\left(-4x\right)+\left(-5\right)\cdot \:5=0\)

\(\Rightarrow x^2x-4xx+5x-5x^2+5\cdot \:4x-5\cdot \:5=0\)

\(\Rightarrow x^3-9x^2+25x-25=0\)

The equation for this polynomial function is \(\rm x^3-9x^2+25x-25=0\) .

The other root of the polynomial is (2- i).

Given

Suppose one of the roots of the polynomial function is complex.

The roots of the function are 2 + i, and 5.

What is the complex conjugate theorem?

The complex conjugate root theorem states that if f(x) is a polynomial with real coefficients and a + ib is one of its roots.

Where a and b are real numbers.

A real coefficient polynomial P(x), if  \(\rm (a+ib)\) is a root then \(\rm (a-ib)\)  will also be a root of P(x).

The roots of the function are \(\rm 5, \ (2+i), \ (2-i)\).

Then,

The equation for this polynomial function is;

\(\rm (x-5)\times (x- (2+i)) \times (x-(2-i))=0\\\\(x-5) \times (x-2-i)\times (x-2+i)=0\\\\(x-5) \times ((x-2)^2-(i)^2=0\\\\(x-5) \times (x^2-4x+5)=0\\\\ x (x^2-4x+5)-5 (x^2-4x+5)=0\\\\x^3-9x^2+25x-25=0\)

Hence, The equation for this polynomial function is\(\rm x^3-9x^2+25x-25=0\).

To know more about complex conjugate theorem click the link given below.

https://brainly.com/question/844178

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