Question

I can’t figure out how to use the zeros in the polynomial. Please explain

Answer 1

a)
`P (x) = (x + 3) (x-1) (x-4)`

b)
`P (x) = (2x + 5) (5x - 4) (x-6)`

c)
`P (x) = (x-3) (x-1) (x-4) (x + 1) ^ 2`

Explanation:

For the question a * you need to find a polynomial of degree 3 with zeros in -3, 1 and 4.

This means that the polynomial P(x) must be zero when x = -3, x = 1 and x = 4.

Then write the polynomial in factored form.

`P (x) = (x + 3) (x-1) (x-4)`

Note that this polynomial has degree 3 and is zero at x = -3, x = 1 and x = 4.

For question b, do the same procedure.

Degree: 3

Zeros: -5/2, 4/5, 6.

The factors are

`x = -(5)/(2)\\\\x +(5)/(2) = 0\\\\(2x +5) = 0`

---------------------------------------

`x =(4)/(5)\\\\x-(4)/(5) = 0\\\\(5x-4) = 0`

--------------------------------------

`x = 6\\\\(x-6) = 0`

--------------------------------------

`P (x) = (2x + 5) (5x - 4) (x-6)`

Finally for the question c we have

Degree: 5

Zeros: -3, 1, 4, -1

Multiplicity 2 in -1

`x = -3\\\\(x-3) = 0`

--------------------------------------

`x = 1\\\\(x-1) = 0`

--------------------------------------

`x = 4\\\\(x-4) = 0`

----------------------------------------

`x = -1\\\\(x + 1) = 0`

-----------------------------------------

`P (x) = (x-3) (x-1) (x-4) (x + 1) ^ 2`