Question

Divide x4-5x2- 4x by x+3 find the remainder by remainder theorem verify by finding the remainder using synthetic division.

Answer 1

Step-by-step explanation:

Quotient and remainder is as given in the picture.

Answer 2

The remainder is 48.

It has been verified by both methods.

Given:

• ${x}^{4} - 5 {x}^{2} - 4x$
• (x+3)

To find:

• find the remainder by remainder theorem.
• Verify by finding the remainder using synthetic division.

Solution:

Theorem to be used:

Factor theorem:

If p(x) is a polynomial and it is divided by (x-a), then remainder is value of polynomial at point a, I.e. p(a).

Step 1:

Find remainder using remainder theorem.

As

$x + 3 = 0 \\ x = - 3$

Let

$p(x) = {x}^{4} - 5 {x}^{2} - 4x$

so,

$p( - 3) = {( - 3)}^{4} - 5 ({ - 3)}^{2} - 4( - 3)$

or

$p( - 3) = 81 - 45 + 12$

or

$p( - 3) = 93 - 45$

or

$p( - 3) = 48$

Remainder is 48.

Step 2:

Perform division by synthetic division.

Write the coefficients of dividend in RHS and write the value of x from divisor in LHS.

Take 1 coefficients down,

Multiply it with -3 and place the value under coefficient of x³,add both and place down,repeat the process till end.

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 1 \: \: \: 0 \: \: \: \: \: -5 \: \: \: \: \: \: -4\: \: \: \: \: \:0\\ x = - 3 \: \: \: \: \: \: \: \: \: \: \: \: -3 \: \: \: \:\: \: \: 9\: \: \: - 12 \: \: \:\:48 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:---------------- \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \:1 \: \: \: \: - 3 \: \: \: \: 4\: \: \: - 16\: \: \:\:\:48$

Thus,

Remainder is 48.

It has been verified by long division also.

Learn more:

1) (x4-x3+2)÷(x+2) polynomial

Long division

synthetic division

reminder theorem

factor theorem

(Solution)

https://brainly.in/question/11569266

2) find the zeros of the cubic polynomial 3 x cube minus 5 x square - 11 x minus 3

https://brainly.in/question/10755381