In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x⁴−2x³−5x²+8x+4=0

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Identify the polynomial given: \(x^{4} - 2x^{3} - 5x^{2} + 8x + 4 = 0\).

For part (a), list all possible rational roots using the Rational Root Theorem. The possible roots are of the form \(\pm \frac{p}{q}\), where \(p\) divides the constant term (4) and \(q\) divides the leading coefficient (1). So, possible roots are \(\pm 1, \pm 2, \pm 4\).

For part (b), test these possible roots by substituting them into the polynomial or by using synthetic division to find which are actual roots. When a root is found, perform polynomial division (synthetic or long division) to divide the polynomial by \((x - \text{root})\) to get a quotient polynomial of degree 3.

Use the quotient polynomial from part (b) and repeat the process: find possible rational roots of the quotient polynomial, test them, and factor further until the polynomial is completely factored or reduced to a quadratic.

For part (c), solve the remaining quadratic (if any) by factoring, completing the square, or using the quadratic formula to find the remaining roots and thus solve the original equation completely.

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주요 개념

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Rational Root Theorem

The Rational Root Theorem helps identify all possible rational roots of a polynomial equation by considering factors of the constant term and the leading coefficient. These possible roots are expressed as ±(factors of constant term)/(factors of leading coefficient), providing a finite list to test.

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide a polynomial by a binomial of the form (x - c). This process simplifies the polynomial after finding one root, reducing its degree and making it easier to find remaining roots. Synthetic division is a shortcut method often used for this purpose.

Solving Polynomial Equations by Factoring

After reducing the polynomial's degree through division, factoring the resulting polynomial or using other root-finding methods helps find all roots. Factoring breaks the polynomial into simpler polynomials whose roots can be found directly, completing the solution of the equation.

In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x4−2x3−5x2+8x+4=0