Find all the zeros of the function and write the polynomial as a product of linear factors. f(x)

step 1 We have the function f of x equals x cubed minus x squared plus x squared plus 39. I'm going to

Find all the zeros of the function and write the polynomial...

Key Concepts

Complex Conjugate Root Theorem

For polynomials with real coefficients, the complex conjugate root theorem guarantees that any non-real complex roots will occur in conjugate pairs. This means that if a complex number a + bi is a root, then its conjugate a - bi is also a root, ensuring that the polynomial can be factored into real linear or quadratic factors.

Zeros of a Polynomial

The zeros (or roots) of a polynomial are the values of the variable for which the polynomial evaluates to zero. Identifying these zeros is crucial for understanding the behavior of the function and for expressing the polynomial as a product of linear factors, where each factor corresponds to a zero.

Factor Theorem

The factor theorem states that if a polynomial f(x) has a zero at x = r (meaning f(r) = 0), then (x - r) is a factor of that polynomial. This theorem is used to break down a polynomial into simpler components, facilitating both the finding of remaining roots and factoring the polynomial completely.

Rational Root Theorem

The rational root theorem provides a tool to list all possible rational zeros of a polynomial with integer coefficients by considering factors of the constant term and the leading coefficient. This theorem offers a systematic approach for testing potential zeros, which can simplify the process of factoring the polynomial.

Polynomial Division

Polynomial division, including synthetic division, is a method used to divide a given polynomial by a suspected linear factor. This technique helps verify if a factor is valid (by resulting in a zero remainder) and reduces the degree of the polynomial, making the process of finding additional roots more manageable.

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