Simplify ³√2⁵×3³×27×2
Answers
Answer:
Step-by-step explanation:
factor theorem says that if x - a is a factor of f(x) if and only if f(a) = 0...
F(x) = x^n + a^n
F(-a) = -(1)^n a^n + a^n = 0 for odd n... as -1)^n = -1..
hence x + a is a factor of F(x)
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F(x) = x⁴ - 4 / x⁴ = (x⁸ - 4) / x⁴
Let a = √√2 = (2)¹/⁴
then a² = √2 and a⁴ = 2 then a⁸ = 4
F(a) = (a⁸ - 4) / a⁴ = 0 => x - a is a factor of F(x)
F(x) = (x - a) (x⁷ + b x⁶ + c x⁵ + d x⁴ + e x³ + f x² + g x + 4/a) / x⁴
= [x⁸ + (b-a) x⁷ + (c-ab) x⁶ + (d-ac) x⁵ + (e-ad) x⁴ + (f - ae) x³
+ (g - af) x² + (4/a - ag) x - 4 ] / x⁴
= (x⁸ - 4) / x⁴
=> b = a
c = ab = a² = √2
d = ac = a³ = 2³/⁴
e = ad =a⁴ = 2
f = ae = a⁵ = 2⁵/⁴
g = a⁶ = 2³/²
4/a = ag = a⁷ => a⁸ = 4.. which i s true,,
F(x) = (x - a) (x⁷ + a x⁶ + √2 x⁵ + a³ x⁴ + 2 x³ + 2⁵/⁴ x² + 2³/² x + 4/a) / x⁴
= (x - a) (x³ + a x² + √2 x + 2³/⁴ + 2 x⁻¹ + 2⁵/⁴ /x⁻² + 2√2 x⁻³ + 4/a x⁻⁴)
Factorizing :
F(x) = (x⁸ - 4) / x⁴ = (x⁴ - 2) (1 + 2 x⁻⁴)
= (x² - √2) (x² + √2) (1 + 2 x⁻⁴)
= (x - a) (x + a) (x² + √2) (a + 2 x⁻⁴)
where a² = √2
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x+2 is a factor of F(x) = (x + 1)⁷ + (2x + k)³
=> F(-2) = 0
=> -1⁷ + (k-4)³ = 0
=> k-4 = 1
=> k = 5
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divisor x^2 + x - 12 = (x + 4) (x - 3)
Let x^3 - 6 x^2 - 15 x + 80
= (x + 4) (x - 3) * (x + a) + [ b (x + 4) + c (x - 3) ]
Thus the remainder after division by divisor is (b + c) x + 4 b - 3 c
Let F(x) = x^3 - 6 x^2 - 15 x + 80 - (b + c) x - 4 b + 3 c
F(-4) = 0 as it should be divisible by x + 4
=> -64 - 96 + 60 + 4 (b+c) - 4 b + 3c = 0
=> c = 100/7
F(3) = 0 as F(x) is divisible by x - 3
=> 27 - 54 - 45 + 80 - 3(b+c) - 4 b + 3 c = 0
=> b = 8/7
So 4 * [ 27 x - 67] /7 must be subtracted from the iven polynomial for the divisor to exatly divide it....