Let fx be a polynomial A basic algebra states that c is a ro

Let f(x) be a polynomial. A basic algebra states that c is a root if f(x) if and only if f(x)= (x-c)g(x) for some polynomial g(x). We say that c is a multiple root if f(x)=(c-x)²h(x) where h(x) is a polynomial.

a) show that c is a multiple root of f(x) if and only if c is a root of both f(x) and f\'(x).

b) use part a to determine whether c= -1 is a multiple root of each of the following polynomials:

(i) y=x⁵+2x⁴-4x³-8x²-x+2

(ii) y= x⁴+x³-5x²-3x+2

Solution

a)

If c is a multipe root of f(x) then

f(x)=(x-c)^2h(x)

f\'(x)=2(x-c)h\'+(x-c)^2h

f\'(c)=0

Hence, a root of f(x) and f\'(x)

Let c be a root of f(x) and f\'(x)

f(x)=(x-c)h(x)

f\'(x)=h(x)+(x-c)h\'(x)

f\'(c)=0=h(c)

Hence, h(c)=0

So, h(x)=(x-c)g(x)

So, f(x)=(x-c)^2g(x)

So,c is a multipel root of f(x)

b)

i)

Setting x=-1 gives

y(-1)=-1+2+4-8+1+2=0

y\'=5x^4+8x^3-12x^2-16x-1

y\'(-1)=5-8-12+16-1=0

Hence, c=-1 is a multiple root

ii)

y(-1)=1-1-5+3+2=0

y\'(x)=4x^3+3x^2-10x-3

y\'(-1)=-4+3+10-3=6

Hence, c=-1 is not a multiple root