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

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

2020-11-09
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)
$$\displaystyle{f{{\left({x}\right)}}}={\left({x}+{4}\right)}{\left({\left({x}-{2}\right)}^{{2}}+{9}\right)}$$
$$\displaystyle{f{{\left({x}\right)}}}={\left({x}+{4}\right)}{\left({x}^{{2}}-{4}{x}+{13}\right)}$$
$$\displaystyle{f{{\left({x}\right)}}}={x}^{{3}}-{3}{x}+{52}$$
Hence, the polynomial is $$\displaystyle{x}^{{3}}-{3}{x}+{52}$$

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