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

A polynomial is a type of expression in which the exponents of all variables should be a whole number.

Answer: The polynomial by which (x⁴ + 5x³ – 3x – 15) has to be divided to get the quotient as (x³ – 3) is (x + 5).

Let's look into the steps below.

Explanation:

Given: Dividend = x⁴ + 5x³ – 3x – 15, Quotient = x³ – 3

To find: Divisor polynomial

The Division algorithm for polynomials says if p(x) and g(x) are the two polynomials, where g(x) ≠ 0, we can write the division of polynomials as:

p(x) = q(x) × g(x) + r(x)

Where,

•  p(x) is the dividend.
•  q(x) is the quotient.
•  g(x) is the divisor.
•  r(x) is the remainder.

Thus, p(x) = x⁴ + 5x³ – 3x – 15, q(x) = x³ – 3, g(x) = ?

r(x) = 0 [ Since the polynomials are exactly divisible to give the quotient (x³ – 3)]

Thus, we can say that 

p(x) = q(x) × g(x) + 0 ------------------ (1)

p(x) / q(x) = g(x) ------------------ (2)

Hence, to calculate the value of g(x) we will divide p(x) by q(x)

Substituting the values of the polynomials in (2) we get,

g(x) = (x⁴ + 5x³ – 3x – 15) / (x³ – 3) 

We will use the long division of polynomials to find the value of g(x)

Let's look into the division shown below:

Thus, we see that

(x⁴ + 5x³ – 3x – 15) / (x³ – 3) = x + 5

Hence, the value of divisor polynomial g(x) = x + 5

Verification:

Substitute the values of q(x) and g(x) in (1)

p(x) = (x³ – 3) (x + 5)

= x⁴ + 5x³ – 3x – 15 

Thus, we see that LHS = RHS

We can also use Cuemath's online polynomial calculator to perform different arithmetic operations on polynomials.

Hence, the polynomial by which (x⁴ + 5x³ – 3x – 15) has to be divided to get the quotient as (x³ – 3) is (x + 5).