# Use the linear factorization theorem to construct a polynomial function with the given condition. n=3 with real coefficients, zeros = -4, 2+3i.

Use the linear factorization theorem to construct a polynomial function with the given condition.
n=3 with real coefficients, zeros = -4, 2+3i.
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Step 1
Given:
f(x) is a polynomial with real coefficients of degree 3
Zeroes of f(x) are -4,2 + 3i
Since f(x) is a polynomial with real coefficients the complex roots exist as complex cojugates
So, 2-3i is a zero of f(x) as 2+3i is a zero.
Step 2
Hence, f(x)=(x-4)(x-(2+3i))(x-(2-3i))
f(x)=(x-(-4))((x-2)-3i)((x-2)+3i)
$f\left(x\right)=\left(x+4\right)\left({\left(x-2\right)}^{2}+9\right)$
$f\left(x\right)=\left(x+4\right)\left({x}^{2}-4x+13\right)$
$f\left(x\right)={x}^{3}-3x+52$
Hence, the polynomial is ${x}^{3}-3x+52$