Studdy Solution
STEP 1
1. The polynomial equation x4+2x3−8x2−18x−9=0 is a fourth-degree equation.
2. The roots of the polynomial can be found using methods such as factoring, synthetic division, or numerical methods if needed.
3. The polynomial may have real and/or complex roots.
STEP 2
1. Perform an initial analysis to identify any possible rational roots using the Rational Root Theorem.
2. Use synthetic division or polynomial division to factor the polynomial, if possible.
3. Solve the resulting lower-degree polynomials (quadratic or cubic equations) to find the remaining roots.
4. Verify all roots by substituting them back into the original polynomial equation.
STEP 3
Apply the Rational Root Theorem to list all possible rational roots of the polynomial.
The Rational Root Theorem states that any possible rational root, p/q, is a factor of the constant term (-9) divided by a factor of the leading coefficient (1).
Possible rational roots:
±1,±3,±9
STEP 4
Test the possible rational roots by substituting them into the polynomial to see if they yield zero.
Testing x=1:
14+2(1)3−8(1)2−18(1)−9=1+2−8−18−9=−32=0
Testing x=−1:
(−1)4+2(−1)3−8(−1)2−18(−1)−9=1−2−8+18−9=0
Thus, x=−1 is a root.
STEP 5
Use synthetic division to divide the polynomial by x+1.
−1112−11−8−1−9−189−9−990
The quotient is x3+x2−9x−9.
STEP 6
Factor the resulting cubic polynomial x3+x2−9x−9 further by rational root testing.
Testing x=1:
13+12−9(1)−9=1+1−9−9=−16=0
Testing x=−1:
(−1)3+(−1)2−9(−1)−9=−1+1+9−9=0
Thus, x=−1 is again a root.
STEP 7
Use synthetic division to divide x3+x2−9x−9 by x+1.
−1111−10−90−9−990
The quotient is x2−9.
STEP 8
Solve the quadratic equation x2−9=0.
x2−9=0⟹(x−3)(x+3)=0
Thus, x=3 and x=−3.
STEP 9
Combine all roots found: x=−1 (twice), x=3, and x=−3.
Verify by substituting back into the original polynomial:
(−1)4+2(−1)3−8(−1)2−18(−1)−9=1−2−8+18−9=0
(3)4+2(3)3−8(3)2−18(3)−9=81+54−72−54−9=0
(−3)4+2(−3)3−8(−3)2−18(−3)−9=81−54−72+54−9=0
All roots satisfy the original polynomial equation.
The roots of the polynomial x4+2x3−8x2−18x−9=0 are:
x=−1,x=−1,x=3,x=−3
