Question

The polynomial of degree 5, P(x) P(x) has leading coefficient 1, has roots of multiplicity 2 at x=5 and x=0, and a root of multiplicity 1 at x=−1

Find a possible formula for P(x)

Answer 1

Since the polynomial has roots of multiplicity 2 at x=5 and x=0, factors of (x−5) and( x=0 ) factors of
`(x-5) ^2`must be present. Also, since it has a root of multiplicity 1 at x=−1, the factor (x+1) must be present.

Since P(x) has a root of multiplicity 2 at x = 5, the factor
`(x - 5)^2` must be present in the polynomial. Similarly, since P(x) has a root of multiplicity 2 at x = 0, the factor
`x^2` must be present. Additionally, since P(x) has a root of multiplicity 1 at x = -1, the factor (x + 1) must be present.

Therefore, a possible formula for P(x) is:

`P(x) = k(x - 5)^2 x^2 (x + 1)`

where k is a constant. Since the leading coefficient of P(x) is 1, we can solve for k by substituting x = 0 and setting P(x) = 1:

`1 = k(0 - 5)^2 0^2 (0 + 1)`

Solving for k, we find k = 1.

Therefore, a possible formula for P(x) is:

`P(x) = (x - 5)^2 x^2 (x + 1)`

Complete question below:

The polynomial of degree 5, P(x), has leading coefficient 1, has roots of multiplicity 2 at x = 5 and x = 0, and a root of multiplicity 1 at x = -1. Find a possible formula for P(x).

Answer 2

`P(x) = x^5 -9x^4 + 10x^3 + 25x^2`

Explanation:

The given degree of the polynomial P(x) = 5

The leading coefficient = 1

So, the general form of the polynomial with degree 5 is
`x^5 + bx^4 + cx^3 + dx^2 + ex + f`

Now root x =5 is of multiplicity 2, x = 0 of multiplicity 2, x = -1 of multiplicity 1

If x = a is the zero of the polynomial of multiplicity m, then ,
`(x-a)^m` is the factor of the polynomial.

⇒
`(x-5)^2` is a factor of P(x)

`(x-0)^2`is another factor of P(x)

(x +1) is the last factor of P(x)

So, P(x) = Product of all factors =
`(x-5)^2 (x)^2(x+1)`

Solving the above expression , we get

`P(x) = (x^2 + 25 -10x) (x^3 + x^2)  = x^3(x^2 + 25 -10x) +x^2(x^2 + 25 -10x) \\= x^5 + 25 x^3 -10x^4 + x^4 +25x^2 -10x^3 \\=x^5 -9x^4 + 10x^3 + 25x^2`

Hence,
`P(x) = x^5 -9x^4 + 10x^3 + 25x^2`