Question

Determine the leading coefficient of the polynomial p, which has exactly three roots: -1, 1 and 2, and for which p(3) = 24 is true.

Answer 1

Answer:

The polynomial is $x^3-2x^2-x+2$

Leading coefficient is 1

Step-by-step explanation:

We need to determine the leading coefficient of polynomial p having roots

-1,1 and 2 and p(3)=24

Roots of polynomial are: -1,1,2

We can write them as:

x=-1, x=1, x=2

x+1=0, x-1=0, x-2=0

or:

(x+1)(x-1)(x-2)=0

Multiplying the terms to find the polynomial:

$(x+1)(x-1)(x-2)=0\\(x(x-1)+1(x-1))(x-2)=0\\(x^2-x+x-1)(x-2)=0\\(x^2-1)(x-2)=0\\x^2(x-2)-1(x-2)=0\\x^3-2x^2-x+2=0$

So, The polynomial is $x^3-2x^2-x+2$

The leading coefficient is the coefficient with highest power So, in polynomial we have highest power $x^3$ so, leading coefficient is 1