Question

Let a(x) = 5x⁹− 8x^5 + 4x³ + 8x and b(x) = x⁴. When dividing a(x) by b(x), what is the unique quotient polynomial q(x) and remainder polynomial r(x) that satisfy the equation?

Answer 1

To divide the polynomials a(x) and b(x), we can use polynomial long division. The quotient polynomial q(x) is 5x^5 and the remainder polynomial r(x) is 0.

Step-by-step explanation:

To divide the polynomial a(x) by b(x), we can use polynomial long division. Divide the leading term of a(x) by the leading term of b(x) to get the first term of the quotient polynomial q(x). In this case, the leading term of a(x) is 5x^9 and the leading term of b(x) is x^4. Dividing these terms gives us q(x) = 5x^5.

Multiply b(x) by q(x) to get the product p(x). Subtract p(x) from a(x) to get the remainder polynomial r(x). In this case, p(x) = 5x^5 * x^4 = 5x^9 and r(x) = a(x) - p(x) = 5x^9 - 5x^9 = 0.

So, the quotient polynomial q(x) is 5x^5 and the remainder polynomial r(x) is 0.

Answer 2

The unique quotient polynomial q(x) = 5x⁵ and the remainder r(x) = 0.

Long division of polynomial function.

The division of polynomial using long method of division involves the use of the divisor to divide each term in the dividend to find the quotient and remainder.

In the given question, we are to divide:

`(a(x) )/(b(x))=(5x^9 -8x^5+4x^3+8x)/(x^4)`

Using long division method:

5x⁵

`x^4` ||
`5x^9 - 8x^5 + 4x^3 + 8x`

`-(5x^9-5x^5)`

`-3x^5+4x^3+8x`

`-(-3x^5)`

`4x^3 +8x`

`-4x^3`

8x

-8x

0

Therefore, we can conclude that the unique quotient polynomial q(x) = 5x⁵ and the remainder r(x) = 0.