Question

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

Accepted Answer

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$$. For part (b), test these possible roots by substituting them into the polynomial or by using synthetic division to find which are actual roots. When a root is found, perform polynomial division (synthetic or long division) to divide the polynomial by (x$$ - 	ext{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.

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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Key Concepts

Rational Root Theorem

Polynomial Division (Synthetic or Long Division)

Solving Polynomial Equations by Factoring