# How do you form a polynomial f(x)with real coefficients having given degree and zeros? Degree 4; zeros -5+2i; 3 multiplicity 2 How do you form a polynomial f(x)with real coefficients having given degree and zeros? Degree 5; zeros:-4; -i; -3+i

Jul 17, 2018

1)$f \left(x\right) = {x}^{4} + 4 {x}^{3} - 22 {x}^{2} - 84 x + 261$
2)$f \left(x\right) = {x}^{5} + 10 {x}^{4} + 3 {x}^{3} + 50 {x}^{2} + 34 x + 40$

#### Explanation:

1)$x = - 5 + 2 i , x = - 5 - 2 i , x = 3 , x = 3$

$f \left(x\right) = \left(x + 5 - 2 i\right) \left(x + 5 + 2 i\right) {\left(x - 3\right)}^{2}$

Let:

$\left(x + 5 - 2 i\right) \left(x + 5 + 2 i\right) = A , {\left(x - 3\right)}^{2} = B$

$A = {x}^{2} + 5 x + 2 i x + 5 x + 25 + 10 i - 2 i x - 10 i - 4 {i}^{2}$

But: $\implies {i}^{2} = - 1$

$\therefore$ $A = {x}^{2} + 10 x + 29$

$B = {x}^{2} - 6 x + 9$

$f \left(x\right) = A \cdot B = \left({x}^{2} + 10 x + 29\right) \left({x}^{2} - 6 x + 9\right)$

$f \left(x\right) = {x}^{4} - 6 {x}^{3} + 9 {x}^{2} + 10 {x}^{3} - 60 {x}^{2} + 90 x + 29 {x}^{2} - 174 x + 261$

$f \left(x\right) = {x}^{4} + 4 {x}^{3} - 22 {x}^{2} - 84 x + 261$

2)$x = - 4 , x = \pm \left(i\right) , x = \left(- 3 \pm i\right)$

$f \left(x\right) = \left(x + 4\right) \left(x - i\right) \left(x + i\right) \left(x + 3 - i\right) \left(x + 3 + i\right)$

$f \left(x\right) = \left(x + 4\right) \left({x}^{2} + 1\right) \left({x}^{2} + 6 x + 10\right)$

$f \left(x\right) = {x}^{5} + 10 {x}^{4} + 3 {x}^{3} + 50 {x}^{2} + 34 x + 40$