without actual division show that 2X2 by 4 minus 6 x raise to power 3 + 3 X raise to power 2 + 3 x minus 2 is exactly divisible by X raise to power 2 - 3 x + 2
Answers
Answer:
Step-by-step explanation:
without actual division, prove that (2x^4-6x^3+3x^3+3x-2)is exactly divisible by(x^2-3x+2).
(note : here'^'this symbol is being referred as raised to.)
The answer:
2x^4 - 6x^3 + 3x^3 + 3x - 2
Combine like terms to simplify
2x^4 - 3x^3 + 3x - 2
Factor the polynomial
2x^4 - 3x^3 + 3x - 2
Use the "RATIONAL ROOT TEST" to find any possible rational roots.
(a_n)x^n + (a_n-1)x^(n-1) + . . . + (a_1)x + a_0
1. find a_n and a_0
a_n (the first coefficient) = 2
a_0 (the constant) = 2
2. Determine factors of a_n and a_0
Factors of a_n = 1, 2
Factors of a_0 = 1, 2
3. Determine possible rational roots
(prelim) Possible rational roots = ±{1/1, ½, 2/1, 2/2}
Eliminate duplicates
(final) Possible rational roots = ±{1, ½, 2}
4. Determine if any of the possible rational roots are actually roots where f(x) = 0
Is x = +1 a rational root ?
f(x) = 2x^4 - 3x^3 + 3x - 2
f(x) = 2*1^4 - 3*1^3 + 3*1 - 2
f(x) = 2 - 3 + 3 - 2
f(x) = 0
YES, x = +1 is a ROOT
Is x = -1 a rational root ?
f(x) = 2x^4 - 3x^3 + 3x - 2
f(x) = 2*(-1)^4 - 3*(-1)^3 + 3*(-1) - 2
f(x) = 2*1 - 3*(-1) + 3*(-1) - 2
f(x) = 2 + 3 - 3 - 2
f(x) = 0
YES, x = -1 is a ROOT