Question

Which polynomial function has a leading coefficient of 1, roots –2 and 7 with multiplicity 1, and root 5 with multiplicity 2?

Answer 1

A polynomial with 'a' as the leading coefficient & b, c & d as roots with multiplicity p, q ,r is written as

$p(x)=a(x-b)^p(x-c)^q(x-d)^r$

Here

a= 1 , b = -2, c = 7, d = 5

p = 1, q=1, r = 2

Substituting the values we get

$p(x)=1(x+2)(x-7)(x-5)^2$

Now

$(x-5)^2=x^2-10x+25$

and

$(x+2)(x-7)=x^2-7x+2x-14=x^2-5x-14$

So

$p(x)=(x^2-5x-14)(x^2-10x+25)$

Now

$(x^2-5x-14)(x^2-10x+25)=x^4-5x^3-14x^2-10x^3+50x^2+140x+25x^2-125x-350$

So

$p(x) = x^4-15x^3+61x^2+15x-350$

Polynomial function has a leading coefficient of 1, roots –2 and 7 with multiplicity 1, and root 5 with multiplicity 2 is

$p(x) = x^4-15x^3+61x^2+15x-350$ ....Answer