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

slaggingV 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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Expert Answer

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(x)=(x(1))(x0)(x(3+i)(x(3i))
=x(x+1)(x3i)(x3+i)
=x(x+i){(x3)2(i)2}
=x(x+1){x26x+9+1}
=(x2+x)(x26x+10)
=x46x3+10x2+x36x2+10x
=x45x3+4x2+10x
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