Answer:
x
4
−5x
2
+4
To factor the expression, solve the equation where it equals to 0.
x
4
−5x
2
+4=0
By Rational Root Theorem, all rational roots of a polynomial are in the form
q
p
, where p divides the constant term 4 and q divides the leading coefficient 1. List all candidates
q
p
.
±4,±2,±1
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
x=1
By Factor theorem, x−k is a factor of the polynomial for each root k. Divide x
4
−5x
2
+4 by x−1 to get x
3
+x
2
−4x−4. To factor the result, solve the equation where it equals to 0.
x
3
+x
2
−4x−4=0
By Rational Root Theorem, all rational roots of a polynomial are in the form
q
p
, where p divides the constant term −4 and q divides the leading coefficient 1. List all candidates
q
p
.
±4,±2,±1
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
x=−1
By Factor theorem, x−k is a factor of the polynomial for each root k. Divide x
3
+x
2
−4x−4 by x+1 to get x
2
−4. To factor the result, solve the equation where it equals to 0.
x
2
−4=0
All equations of the form ax
2
+bx+c=0 can be solved using the quadratic formula:
2a
−b±
b
2
−4ac
. Substitute 1 for a, 0 for b, and −4 for c in the quadratic formula.
x=
2
0±
0
2
−4×1(−4)
Do the calculations.
x=
2
0±4
Solve the equation x
2
−4=0 when ± is plus and when ± is minus.
x=−2
x=2
Rewrite the factored expression using the obtained roots.
(x−2)(x−1)(x+1)(x+2)