Question

F(x)=x^(3)-6x^(2)+21x-26 use the complex zeros to factor

Answer 1

The factored form of
`\(f(x) = x^3 - 6x^2 + 21x - 26\) using its complex zeros is \((x - (1 + 2i))(x - (1 - 2i))(x - 3)\).`

Step-by-step explanation:

The given cubic function
`\(f(x) = x^3 - 6x^2 + 21x - 26\)` has complex zeros. Complex zeros occur in conjugate pairs for real-coefficient polynomials. In this case, the complex zeros are \(1 + 2i\) and \(1 - 2i\).

Complex conjugate pairs can be expressed as factors in the form \((x - (a + bi))(x - (a - bi))\), where \(a\) and \(b\) are the real and imaginary parts of the complex zero, respectively. Therefore, the factors corresponding to the complex zeros are \((x - (1 + 2i))(x - (1 - 2i))\).

The third zero is a real root, which is \(x = 3\). The factor corresponding to this real zero is \((x - 3)\).

Multiplying these three factors together gives the factored form of the cubic polynomial. The final factored form is \
`/((x - (1 + 2i))(x - (1 - 2i))(x - 3)\),`representing the given cubic function in terms of its complex and real zeros. This form provides a clear understanding of how the roots contribute to the polynomial's behavior.