Can you help me to find the roots of 4x^3-21x^2+12x+32? with each step.?

= 4x³ - 21x² + 12x + 32

= 4x³ - (16x² + 5x²) + (20x - 8x) + 32

= 4x³ - 16x² - 5x² + 20x - 8x + 32

= 4x³ - 5x² - 8x - 16x² + 20x + 32

= (4x³ - 5x² - 8x) - (16x² - 20x - 32)

= x(4x² - 5x - 8) - 4(4x² - 5x - 8)

= (x - 4)(4x² - 5x - 8) ← this is the first root

Let's look the polynomial: 4x² - 5x - 8

Polynomial like: ax² + bx + c, where:

a = 4

b = - 5

c = - 8

Δ = b² - 4ac (discriminant)

Δ = (- 5)² - 4(4 * - 8) = 25 + 128 = 153 = 9 * 17

x1 = (- b - √Δ) / 2a = (5 - 3√17)/8 ← this is the second root

x2 = (- b + √Δ) / 2a = (5 + 3√17)/8 ← this is the third root

Hello,

4𝑥³ – 21𝑥² + 12𝑥 + 32

   = 4𝑥³ – 16𝑥² – 5𝑥² + 20𝑥 – 8𝑥 + 32   ←←← Split -21𝑥² and 12𝑥

   = 4𝑥²(𝑥 – 4) – 5𝑥(𝑥 – 4) – 8(𝑥 – 4)

   = (𝑥 – 4)(4𝑥² – 5𝑥 – 8)

   = (𝑥 – 4)[(2𝑥)² – 2×(2𝑥)×(5/4) + (5/4)² – 25/16 – 128/16] ←←← Complete the square

   = (𝑥 – 4)[(2𝑥 – 5/4)² – (3√17)²/4²] ←←← Because a²–2ab+b²=(a–b)²

   = (𝑥 – 4)[2𝑥 – 5/4 – 3(√17)/4][2𝑥 – 5/4 + 3(√17)/4)]

           ←←← Because a²–b²=(a–b)(a+b)

   = (𝑥 – 4)[2𝑥 – (5 + 3√17)/4][2𝑥 – (5 – 3√17)/4]

Thus the roots, using null factor law:

   𝑥₁ = 4

   𝑥₂ = (5 + 3√17)/8

   𝑥₃ = (5 – 3√17)/8

Regards,

Dragon.Jade :-)

Note:

♠ Splitting -21𝑥²=-16𝑥²–5𝑥² and 12𝑥=20𝑥–8𝑥 is done because we already know that 𝑥–4 will be a factor of the cubic polynomial you wanted solving.

♠ How can we know? Well, through the use of Rational Root Theorem (see link below), we know that some values, namely ±32, ±16, ±8, ±4, ±2, ±1, ±½ and ±¼ have the highest possibilities of being roots.

Finding 4 as root is then the matter of checking by replacing 𝑥 by 4 in the polynomial...

♠ Once assured that 4 is root, we deduced that monomial 𝑥–4 will be factor. Using long division or polynomial divison we'll then factor out 𝑥–4.

♠ Splitting -21𝑥²=-16𝑥²–5𝑥² is then done on purpose.

Since we have 4𝑥³, the only way to factor out 𝑥–4 is to have a following -16𝑥² because:

   4𝑥³ – 16𝑥² = 4𝑥²(𝑥 – 4)

Since we already have -21𝑥² we have splitted it in -16𝑥² (that we desired) and the remainder -5𝑥².

♠ By same logic, splitting 12𝑥 into 20𝑥–8𝑥 is done because we have a remainder above of -5𝑥².

Because:

   -5𝑥² + 20𝑥 = -5𝑥(𝑥 – 4)

we had to split the existing 12𝑥 into 20𝑥 that we wanted with a remainder of -8𝑥.

♠ That last remainder obviously matched the +32 constant to form the last factor:

   -8𝑥 + 32 = -8(𝑥 – 4)

but this comes with no surprise since we knew beforehand that the factoring would be perfect.

♠ Factoring out 𝑥–4 will leave us with a quadratic that can be easily solve by any means you know (I used "complete the square" while la_console used "quadratic formula").