# 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

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
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
$f\left(x\right)=\left(x-\left(-1\right)\right)\left(x-0\right)\left(x-\left(3+i\right)\left(x-\left(3-i\right)\right)$
$=x\left(x+1\right)\left(x-3-i\right)\left(x-3+i\right)$
$=x\left(x+i\right)\left\{{\left(x-3\right)}^{2}-{\left(i\right)}^{2}\right\}$
$=x\left(x+1\right)\left\{{x}^{2}-6x+9+1\right\}$
$=\left({x}^{2}+x\right)\left({x}^{2}-6x+10\right)$
$={x}^{4}-6{x}^{3}+10{x}^{2}+{x}^{3}-6{x}^{2}+10x$
$={x}^{4}-5{x}^{3}+4{x}^{2}+10x$