Find all zeros of f(x) = x³ + 5x² – 8x + 2.

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Start by identifying the polynomial function: \(f(x) = x^{3} + 5x^{2} - 8x + 2\).

Use the Rational Root Theorem to list all possible rational zeros. These are of the form \(\pm \frac{p}{q}\), where \(p\) divides the constant term (2) and \(q\) divides the leading coefficient (1). So possible rational roots are \(\pm 1, \pm 2\).

Test each possible rational root by substituting into \(f(x)\) to see if it equals zero. For example, evaluate \(f(1)\), \(f(-1)\), \(f(2)\), and \(f(-2)\).

Once a root \(r\) is found, use polynomial division or synthetic division to divide \(f(x)\) by \((x - r)\), reducing the cubic to a quadratic.

Solve the resulting quadratic equation using factoring, completing the square, or the quadratic formula to find the remaining zeros.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Zeros

Zeros of a polynomial are the values of x for which the polynomial equals zero. Finding zeros involves solving the equation f(x) = 0, which helps identify the roots or x-intercepts of the polynomial function.

Rational Root Theorem

The Rational Root Theorem provides possible rational zeros of a polynomial by considering factors of the constant term and the leading coefficient. It helps narrow down candidates to test when searching for roots of polynomials with integer coefficients.

Polynomial Division or Synthetic Division

Polynomial or synthetic division is used to divide a polynomial by a binomial of the form (x - c). It helps simplify the polynomial after finding one root, reducing the degree and making it easier to find remaining zeros.

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