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.

2 Answers | Add Yours

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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

Top Answer

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.

We’ve answered 318,944 questions. We can answer yours, too.

Ask a question