Question

HELPPPP PLEAAAASE I HAVE NO IDEA WHAT IM SUPPOSED TO DOOOOO

Answer 1

`\mathrm{Factoring}\:4x^3-15x^2-31x+30:\quad \boxed{\left(x+2\right)\left(4x-3\right)\left(x-5\right)}`

Explanation:

This is a particularly tough question. It involves polynomial division(a real laborious process) and solving using quadratic formula.

I presume your school has provided you the resources for both.

Anyways, here goes a brief explanation

To find the roots(zeros) of a polynomial,
`f(x)` , set
`f(x) = 0` and solve for values of
`x`

In general, if
`a` is a zero of
`f(x)`, then
`f(x) /(x-a)` will leave no remainder. In other words, it is a factor of the polynomial. It also means
`x - a = 0`

The given polynomial is
`4 x^(3) - 15 x^(2) - 31 x + 30`

We are given that
`- 2` is a zero of
`f(x)`. This simply means that
`f(-2) = 0`

Plugging
`-2` into the function gives

`f(-2) = 4(-2)^3 -15(-2)^2 - 31x + 30\\\\ = 0 =4\left(-8\right)-15\cdot \:4-31\left(-2\right)+30`

Which indeed is 0

Here
`a = -2` , so
`x - (-2)` which is
`x + 2` is a factor of
`f(x)`. It also means that

`x-(-2) = 0 \;or\; x + 2 = 0`

Steps to factor the polynomial
`4 x^(3) - 15 x^(2) - 31 x + 30` :

• Divide
  `4 x^(3) - 15 x^(2) - 31 x + 30` by
  `x+2`
  
  
  `(4 x^(3) - 15 x^(2) - 31 x + 30)/(x + 2) = 4x^2-23x\:+\:15`

• The quotient is a quadratic equation which can be solved using the quadratic formula available in most scientific calculators
  The roots of this quadratic equation are
  
  `x=5,\:x=(3)/(4)`
  
• This means
  `x - 5` is one factor and
  
  `x - (3)/(4)`
  is another factor
  
• Since
  `(3)/(4)` is a zero,
  `x - (3)/(4) = 0`
  

Multiplying both sides of the above by
`4` gives
`4x -3 = 0.` So
`4x-3` is the third factor, replacing
`x - (3)/(4)`

The factors of
`4 x^(3) - 15 x^(2) - 31 x + 30`
are:

`x+2, x-5 \;and \;4x-3`

Answer:

`\mathrm{Factor}\:4x^3-15x^2-31x+30:\quad \boxed{\left(x+2\right)\left(4x-3\right)\left(x-5\right)}`

Use a polynomial division calculator which you can find on the web and a quadratic formula calculator (on the we or on your own calculator)