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How do you use the remainder theorem and Synthetic Division to find the remainders in the following division problems #-2x^4 - 6x^2 + 3x + 1 # divided by x+1?

Answer 1

Let #f(x) = -2x^4-6x^2+3x+1#.

Using the remainder theorem, the remainder is

#f(-1) = -2-6-3+1 = -10#

Using Synthetic division we get the same remainder.

The remainder theorem states that the remainder of dividing a polynomial #f(x)# by #(x-a)# is #f(a)#. In our case #a=-1# and #f(-1) = -10#.

Alternatively, using synthetic division we get the same remainder...

Here we divide #-2x^4-6x^2+3x+1# by #x+1#. Note the #0# in #-2, 0, -6, 3, 1#, standing for the missing #x^3# term's coefficient.

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Answer 2

Chloe Alexander

To use the remainder theorem and synthetic division to find the remainder when (-2x^4 - 6x^2 + 3x + 1) is divided by (x+1), follow these steps:

Identify the divisor (x+1) and set it equal to zero to find the root: [ x+1 = 0 ] [ x = -1 ]
Use synthetic division to divide the polynomial by the root (-1):
- Write down the coefficients of the polynomial in descending order of powers of x, including placeholders for missing terms.
- Perform synthetic division using the root (-1) as the divisor.
- The remainder will be the last number in the bottom row.

The synthetic division table looks like this:

[ \begin{array}{c|cccc} -1 & -2 & 0 & -6 & 3 & 1 \ \hline & & -2 & 2 & -4 & 7 \ \end{array} ]

Interpret the result:
- The remainder is 7.

Therefore, when (-2x^4 - 6x^2 + 3x + 1) is divided by (x+1), the remainder is 7.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.