Question

Can someone please help with this one? "write a polynomial equation of degree 4 with leading coefficient 1 that has roots at -2, -1, 3, and 4."?

Answer 1

`(x+2)(x+1)(x-3)(x-4)`

Explanation:

This would give `x^4` as your largest power, expand the brackets to write it as an equation.

Answer 2

`x^4-4x^3-7x^2+22x+24`

Explanation:

Expand `(x+2)(x+1)(x-3)(x-4)`

`(x+2)(x+1)=x^2+x+2x+2=x^2+3x+2`

`(x^2+3x+2)(x-3)=x^3-3x^2+3x^2-9x+2x-6`

=`x^3-7x-6`

`(x^3-7x-6)(x-4)=x^4-4x^3-7x^2+28x-6x+24`

=`x^4-4x^3-7x^2+22x+24`

Answer 3

The polynomial is `x^4-4x^3-7x^2+22x+24=0`

Explanation:

A polynomial equation of degree `4` with leading coefficient `a` and four roots `alpha,beta,gamma,delta` is

`a(x-alpha)(x-beta)(x-gamma)(x-delta)=0`

Hence, a polynomial equation of degree `4` with leading coefficient `1` and four roots `-2,-1,3,4` is

`1(x-(-2))(x-(-1))(x-3)(x-4)=0`

or `(x+2)(x+1)(x-3)(x-4)=0`

or `(x^2+3x+2)(x^2-7x+12)=0`

or `x^4-7x^3+3x^3+12x^2+2x^2-21x^2-14x+36x+24=0`

or `x^4-4x^3-7x^2+22x+24=0`

Answer 4

`f(x) = (x+2)(x+1)(x-3)(x-4)`

Explanation:

You can answer the question by looking at the requirements and fulfilling them step by step. A degree 4 polynomial will have `x^4` in it somewhere, and 4 will be the largest number x is raised to in the equation. For example, `f(x) = x^4 + x^3 + x^2 + x + 1` is an equation with degree 4, since 4 is the largest number `x` is raised to in the equation.

A leading coefficient of 1 means that the highest order term (`x^4` in this case) has the coefficient of 1. For example, `f(x) = 2x^4 + x` has a leading coefficient of 2. You can think of `f(x) = x^4` as having a "secret" 1 infront of the `x^4` term. (i.e. `f(x) = 1x^4`)

Lastly, having roots at -2, -1, 3, and 4 can be fulfilled by writing the polynomial in a way where it is easy to identify roots. A root of the equation is a value of `x` where `f(x) = 0`. So for the equation `f(x)=(x-5)` the root is 5. Again, the roots for the equation `f(x)=(x-2)(x-3)` are 2 and 3, since we have `f(2) = 0` and `f(3) = 0`.

So all put together we can choose the equation `f(x) = (x+2)(x+1)(x-3)(x-4)` which has roots at -2, -1, 3, and 4. We can expand the equation to `f(x) = x^4 - 4 x^3 - 7 x^2 + 22 x + 24` to double check that the equation has a leading coefficient of 1.