A polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express f (x) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R.

$-1,0,3+i.$ degree 4

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2021-09-03
Answered

A polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express f (x) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R.

$-1,0,3+i.$ degree 4

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Viktor Wiley

Answered 2021-09-04
Author has **84** answers

Given Roots of the polynomial of degree 4 is -1, 0, 3+i

To find the polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Also, to Express f (x) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R.

Definition Used We know if a polynomial having a complex root then its complex conjugate will also be the root of the given polynomial.

Formation of the polynomial f(x)

Since by the definition if 3+i is one root then 3-i will also be the root of the given polynomial

In this way, we have a total number of 4 zeroes i.e.

-1,0,3+i and 3-i

Then the polynomial f(x) is

$f\left(x\right)=(x-(-1))(x-0)(x-(3+i)(x-(3-i))$

$=x(x+1)(x-3-i)(x-3+i)$

$=x(x+i)\{{(x-3)}^{2}-{\left(i\right)}^{2}\}$

$=x(x+1)\{{x}^{2}-6x+9+1\}$

$=({x}^{2}+x)({x}^{2}-6x+10)$

$={x}^{4}-6{x}^{3}+10{x}^{2}+{x}^{3}-6{x}^{2}+10x$

$={x}^{4}-5{x}^{3}+4{x}^{2}+10x$

To find the polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Also, to Express f (x) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R.

Definition Used We know if a polynomial having a complex root then its complex conjugate will also be the root of the given polynomial.

Formation of the polynomial f(x)

Since by the definition if 3+i is one root then 3-i will also be the root of the given polynomial

In this way, we have a total number of 4 zeroes i.e.

-1,0,3+i and 3-i

Then the polynomial f(x) is

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