x3 + 12x2 - 5x = x(x2 + 12x - 5) = x(x + 6 - √41)(x + 6 + √41)
(x + 4) / (x3 - 11x + 20) = (x + 4) / (x2 + 4x - 5)(x + 4) = 1 / (x2 + 4x - 5) = 1 / (x + 5)(x - 1), where x ≠ -4
(x - 3)*(x + 5)*(x - 1) = 0 (x2 - 3x + 5x - 15)*(x - 1) = 0 (x2 + 2x - 15)*(x - 1) = 0 (x3 + 2x2 - 15x - x2 - 2x + 15) = 0 ie x3 + x2 - 17x + 15 = 0
x3 + 5x2 - x - 5 = (x2 - 1)(x + 5) = (x + 1)(x - 1)(x + 5)
x2 - 5x = 0x (x - 5) = 0x = 0andx = 5
(x - 1)(x^2 - 5)
x3 + 12x2 - 5x = x(x2 + 12x - 5) = x(x + 6 - √41)(x + 6 + √41)
x3 - 4x2 + 5x - 20 = x2*(x - 4) + 5*(x - 4) = (x2 + 5)*(x - 4)
x2 + 5x + 25
x3 + 125 Use the sum of two cubes. (x+5)(x2-5x+25)
5(x2 + x + 1)
Difference of two cubes: a3 - b3 = (a-b)(a2+ab+b2) 125 - x3 = 53 - x3 = (5 - x)(52 + 5x + x2) = (5 - x)(25 + 5x + x2)
x(x+5)
x3 - 6x2 + 7x - 2 = x3 - x2 - 5x2 + 5x + 2x - 2 = x2(x - 1) - 5x(x - 1) + 2(x - 1) = (x - 1)(x2 - 5x + 2) = (x - 1){x - 0.5*[5 - sqrt(25 - 8)]}){x + 0.5*[5 - sqrt(25 - 8)]} = (x - 1)(x - 0.4384)(x - 4.5616) so that the roots are x = 1 x = 0.4384 and x = 4.5616
No, it is not.
Since the polynomial has three zeroes it has a degree of 3. So we have:(x - -5)[x - (2 + i)][x - (2 - i)]= (x + 5)[(x - 2) - i][(x - 2) + i]= (x + 5)[(x - 2)2 - i2]= (x + 5)[x2 - 2(x)(2) + 22 - (-1)]= (x + 5)(x2 - 4x + 4 + 1)= (x + 5)(x2 - 4x + 5)= x(x2 - 4x + 5) + 5(x2 - 4x + 5)= x3 - 4x2 + 5x + 5x2 - 20x + 25= x3 + x2 - 15x + 25Thus,P(x) = x3 + x2 - 15x + 25
To solve for x: x2 = 5x + 2 x2 - 5x = 2 x2 - 5x + (5/2)2 = 2 + (5/2)2 (x - 5/2)2 = 8/4 + 25/4 x - 5/2 = ± √(33/4) x = (5 ± √33) / 2