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Find a polynomial with integer coefficients that satisfies the given conditions

– The degree of $ Q $ should be $ 3, space 0 $ and $ i $.

The main objective of this question is to find the polynomial for the given conditions.

This question uses the concept of the complex conjugate theorem. According to the conjugate root theorem, if a polynomial for one variable has real coefficients and also the complex number which is $ a + bi $ is one of its roots, then its complex conjugate, a – bi, is also one of its roots.

Expert Answer

We have to find the polynomial for the given conditions.

From the complex conjugate theorem, we know that if the polynomial $ Q (  x ) $ has real coefficients and $ i $  is a zero, it’s conjugate “-i” is also a zero of  $ Q ( x ) $.

Thus:

  • The expression  $  (x – 0) $ is indeed a factor of $ Q $ if $ 0 $ is indeed a zero of $ Q (x) $.
  • The expression  $  (x – 0) $ is indeed a factor of $ Q $ if $ i $ is indeed a zero of $ Q (x) $.
  • The expression  $  (x – 0) $ is indeed a factor of $ Q $ if $  -i $ is indeed a zero of $ Q (x) $.

The polynomial is:

\[ \space Q ( x ) \space = \space ( x \space – \space 0 ) ( x \space – \space i) (x \space + \space 0) \]

We know that:

\[ \space a^2 \space – \space b^2 \space = \space ( a \space + \space b ) ( a \space – \space b ) \]

Thus:

\[ \space Q ( x ) \space = \space x ( x^2 \space – \space i^2 ) \]

\[ \space Q ( x ) \space = \space x ( x^2 \space + \space 1 ) \]

\[ \space Q ( x ) \space = \space x^3 \space + \space x \]

Numerical Answer

The polynomial for the given condition is:

\[ \space Q ( x ) \space = \space x^3 \space + \space x \]

Example

Find the polynomial which has a degree of $ 2 $ and zeros $ 1 \space +  \space i $ with $ 1 \space –  \space i  $.

We have to find the polynomial for the given conditions.

From the complex conjugate theorem, we know that if the polynomial $ Q (  x ) $ has real coefficients and $ i $ is a zero, it’s conjugate “-i” is also a zero of  $ Q ( x ) $.

Thus:

\[ \space ( x \space – \space (1 \space + i)) ( x \space – \space (1 \space –  \space i )) \]

Then:

\[ \space (x \space –  \space 1)^2 \space – \space (i)^2  \]

\[ \space x^2 \space – \space 2 x \space + \space 1 \space – \space ( – 1 ) \]

\[ \space x^2 \space – \space 2 x \space + \space 2 \]

The required polynomial for the given condition is:

\[ \space x^2 \space – \space 2 x \space + \space 2 \]

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