To solve equations with a variable expression raised to a rational exponent, we can raise each side of the equation to the power of the denominator of the rational exponent.
xnm=k⇔(xnm)n=kn
Since nm⋅n=m, this will eliminate the root. In our case, we will first isolate the variable term and raise each side of the equation to the power of 3.
We now have a polynomial, and we need to find its roots. The Rational Root Theorem can help us narrow down the possible roots. Let Q(x)=anxn+an−1xn−1+⋯+a1x+a0 be a polynomial with integer coefficients. There are a limited number of possible roots for Q(x)=0.
Integer Roots
Integer roots must be factors of a0. The constant term for our polynomial is -352. Its factors, and the possible integer roots, are ±1,±2,±4,±8,±11,±16, and ±32. Let's check them.
x
x3−22x2+153x−352
P(x)=x3−22x2+153x−352
1
(1)3−22(1)2+153(1)−352
-220×
-1
(-1)3−22(-1)2+153(-1)−352
-528×
2
(2)3−22(2)2+153(2)−352
-126×
-2
(-2)3−22(-2)2+153(-2)−352
-754×
4
(4)3−22(4)2+153(4)−352
-28×
-4
(-4)3−22(-4)2+153(-4)−352
-1380×
8
(8)3−22(8)2+153(8)−352
-24×
-8
(-8)3−22(-8)2+153(-8)−352
-3496×
11
(11)3−22(11)2+153(11)−352
0✓
-11
(-11)3−22(-11)2+153(-11)−352
-6028×
16
(16)3−22(16)2+153(16)−352
560×
-16
(-16)3−22(-16)2+153(-16)−352
-12528×
32
(32)3−22(32)2+153(32)−352
14784×
-32
(-32)3−22(-32)2+153(-32)−352
-60544×
We can see above that 11 is a root for the polynomial. Let's now try to find rational roots.
Other Possible Roots
Since 11 is a solution for the polynomial, (x−11) is factor of the polynomial. Let's use synthetic division to factor out (x−11). Then we can find other roots using the remaining polynomial.
111-22153-352
Bring down the first coefficient
111-22153-3521-352153-22
Multiply the coefficient by the divisor
111-22-11153-3521-352153-22
Add down
111-22-11153-3521-11-352153
▼
Repeat the process for all the coefficients
Multiply the coefficient by the divisor
111-22-11-153-121-3521-11-352-153
Add down
111-22-11-153-121-3521-11-132-352
Multiply the coefficient by the divisor
111-22-11-153-121-352-3521-11-132-352
Add down
111-22-11-153-121-352-3521-11-132-350
Using synthetic division to remove the first root left us with the following polynomial.
x2−11x+32
Factoring the Remaining Quadratic Factor
We will use the Quadratic Formula to find the remaining factors. To do so, we will need to identify the values of a,b, and c.
1x2−11x+32=0
We can see above that a=1,b=-11, and c=32. Now, let's substitute these values into the Quadratic Formula to find the remaining roots.
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