Question

Given the equation x 4−2x3−10x 2+18x+9=0, complete the following. a. List all possible rational roots. b. Use synthetic division to test several possible rational roots in order to identify one actual root. c. Use the root from part (b) to solve the equation. a. List all rational roots that are possible according to the Rational Zero Theorem. (Use commas to separate answers as needed.) b. Use synthetic division to test several possible rational roots in order to identify one actual root. One rational root of the given equation is (Simplify your answer.) c. Use the root from part (b) to solve the equation. The solution set is . (Simplify your answer. Type an exact answer, using radicals as needed. USe integers or fractions for any numbers in the expression. Use commas to separate answers as needed.)

Answer 1

First, determine all possible rational roots using the Rational Root Theorem. Then, use synthetic division to test these roots to find an actual root. Finally, use the identified root to factor the polynomial and solve the equation.

Step-by-step explanation:

To solve the polynomial equation x4 - 2x3 - 10x2 + 18x + 9 = 0, we will begin by listing all possible rational roots using the Rational Root Theorem, then use synthetic division to test these possibilities and finally solve the equation using the identified rational root.

Part A

According to the Rational Root Theorem, the possible rational roots are the divisors of the constant term (9) divided by the divisors of the leading coefficient (1). Thus, the possible rational roots are ±1, ±3, ±9.

Part B

To find an actual root, we will perform synthetic division using the possible roots. If the remainder is zero, we have found a root.

Part C

After identifying a root, suppose it's 'r', we will factor the equation by dividing it by (x - r). We can then solve the resulting cubic equation using different methods such as synthetic division again, factoring, or numerical approaches. For a quadratic equation part that might result, we use the quadratic formula x = (-b ± sqrt(b2 - 4ac)) / (2a) to find the remaining roots.