5x2 + 20 = 5*(x2 + 4) which cannot be factorised further.
If you notice, 8x + 4 = 4(2x + 1), but there is an odd coefficient for x2, so there is guaranteed to be some remainder (that is 8x + 4 is not a factor of the polynomial): (24x3 - 5x2 - 48x - 8) ÷ (8x + 4) = 3x2 - 2x - 5 - (x2 - 12)/(8x + 4)
If you mean: f(x) = x4 - 3x3 + 5x2 / x2 then: f(x) = x4 - 3x3 + 5 ∴ f'(x) = 4x3 - 9x2 If you mean: f(x) = (x4 - 3x3 + 5x2) / x2 then: f(x) = x2 - 3x + 5 ∴ f'(x) = 2x - 3
To factorise x3 + 5x2 - 16x - 80 I note that -80 = 5 x -16 and 5 & -16 are the coefficients of x2 and x. Thus I have: x3 + 5x2 - 16x - 80 = (x + 5)(x2 - 16) and the second term is a difference of 2 squares, meaning I have: x3 + 5x2 - 16x - 80 = (x + 5)(x + 4)(x - 4)
25
5x2 = 3 then x2 = 3/5 so that x = sqrt(3/5) = ± 0.7746
5x2 - 136 = 44 5x2 - 136 + 136 = 44 + 136 5x2 = 180 5x2/5 = 180/5 x2 = 36 √x2 = √36 x = ±6
To factor any term, divide by the LCF (largest common factor) of the term. For 5x2-45, the LCF is 5. 5x2-45 = 5(x2-9) It can, of course, be factored even further to become: 5(x2-9) = 45(x2/9 - 1) but when factoring, we usually don't need to go past 5(x2-9) (for example, you've figured out that x2 = 9, thus x = +/- 3.)
5(x2 - 25) 5(x + 5)(x - 5)
5x2+500 = 844.45 5x2 = 844.45-500 5x2 = 344.45 x2 = 344.45/5 x2 = 68.89 x = the square root of 68.89 = 8.3 x = 8.3
5x2 - 20xy + 10y2 = 5(x2 - 4xy + 2y2) which cannot be simplified into rational factors.
x3 + 5x2 - x - 5 = (x2 - 1)(x + 5) = (x + 1)(x - 1)(x + 5)