Answer:
(x−3)(x+3)(x^2-2x+5)
Step-by-step explanation:
f(x) = x^4 - 2x^3 - 4x^2 + 18x - 45
To factor the expression, solve the equation where it equals to 0.
x^4 - 2x^3 - 4x^2 + 18x - 45 = 0
By Rational Root Theorem, all rational roots of a polynomial are in the form p/q where p divides the constant term −45 and q divides the leading coefficient 1. List all candidates p/q.
±45,±15,±9,±5,±3,±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=3
By Factor theorem, x−k is a factor of the polynomial for each root k. Divide x^4 - 2x^3 - 4x^2 + 18x - 45 by x−3 to get x^3 + x^2 - x + 15. To factor the result, solve the equation where it equals to 0.
x^3 + x^2 - x + 15 = 0
By Rational Root Theorem, all rational roots of a polynomial are in the form p/q where p divides the constant term 15 and q divides the leading coefficient 1. List all candidates p/q.
±15,±5,±3,±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=−3
By Factor theorem, x−k is a factor of the polynomial for each root k. Divide x^3 + x^2 - x + 15 by x+3 to get x^2 - 2x + 5. To factor the result, solve the equation where it equals to 0.
x^2−2x+5=0
All equations of the form ax^2 + bx + c=0 can be solved using the quadratic formula: −b±√b2-4ac/2a. Substitute 1 for a, −2 for b, and 5 for c in the quadratic formula.
x=-(-2)+√(-2)^2 - 4 x 1 x 5 / 2
Do the calculations.
x=2+√-16 / 2
Polynomial x^2−2x+5 is not factored since it does not have any rational roots.
x^2−2x+5
Rewrite the factored expression using the obtained roots.
(x−3)(x+3)(x^2-2x+5)
