What is the polynomial with highest degree 3 such that when the polynomial is divided by binomials x+2 and x^2-1, the remainder is 3.

Expert Answers
justaguide eNotes educator| Certified Educator

The required ploynomial has a highest degree of 3.

When it is divided by (x + 2) and (x^2 - 1) the remainder is 3.

So we get the polynomial as (x + 2)(x^2 - 1) + 3

=> x^3 + 2x^2 - x - 2 + 3

=> x^3 + 2x^2 - x + 1

The required polynomial is x^3 + 2x^2 - x + 1

giorgiana1976 | Student

The polynomial is not determined, yet. It will be determined, when all it's coefficients will be known.

We'll write the polynomial whose leading term is ax^3 as:

P = ax^3 + bx^2 + cx + d

Since we know the reminder that we've get when P is divided by given binomials, we'll apply the reminder theorem.

We notice that x^2 - 1 is a difference of 2 squares and it represents the product:

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

We'll write the reminder theorem, when P(x) is divided by (X-1):

P(1)=3

We'll write the reminder theorem, when P(x) is divided by (X+1):

P(-1)=3

We'll write the reminder theorem, when P(x) is divided by (X+2):

P(-2)=3

From these facts, we notice that the reminder of the division of   P(x) to the product of polynomials (X-1)(X+1)(X+2) is also 3.

We'll write the reminder theorem:

aX^3+bX^2+cX+d=(X-1)(X+1)(X+2) + 3

We'll remove the brackets:

aX^3+bX^2+cX+d=(X^2-1)(X+2) +3

aX^3+bX^2+cX+d = X^3 + 2X^2-X -2 + 3

We'll combine like terms and we'll get:

aX^3 + bX^2 + cX + d = X^3 + 2X^2 - X + 1

The requested polynomial is: P(X) = X^3 + 2X^2 - X + 1.