Question

The quotient of (x4 + 5x3 – 3x – 15) and a polynomial is (x3 – 3). What is the polynomial?

Answer 1

The polynomial is x+5.

Explanation:

Here we need to recall the algorithm of the division for polynomials, which is very similar to the one for integers. Given polynomials P(x) and Q(x), there always exist polynomials S(x) and R(x) such that

P(x) = Q(x)S(x) + R(x)

where

• R(x) is the remainder,
• S(x) is the quotient,
• Q(x) is the divisor,
• P(x) is the dividend.

In this particular case,

• P(x) = x⁴+5x³-3x-15,
• S(x) = x³-3
• R(x) = 0
• Q(x), is what we are looking for.

Then, x⁴+5x³-3x-15 = (x³-3)Q(x). In order to find Q(x) we must complete the division x⁴+5x³-3x-15/x³-3. This gives us that Q(x)=x+5.

Answer 2

x+5

Explanation:

Given:

Dividend= x4 + 5x3 – 3x – 15

Quotient=(x3-3)

As per the rule of division of polynomial:

Dividend = Divisor × Quotient + Remainder

Divisor= required polynomial p(x)

remainder=0

x4 + 5x3 – 3x – 15= p(x) *(x3-3) + 0

p(x)=x4 + 5x3 – 3x – 15/x3-3

By long division we get

p(x)= x+5 !