Question

What is the remainder? Equation is below.

Answer 1

-23. In my explanation I will include in my picture how this will look in your final answer

Explanation:

So to solve this, I first set x + 3 = 0. This means that x = -3, which we will use soon. Now, here's how you would work out this problem. It would be confusing if I explained over text, so I included a picture of my work.

You would first set up your problem like it is in the picture. Then, bring 2 down. Next, multiply 2 by -3 (for future problems, you would multiply the number you brought down by whatever number is on the side). -3 × 2 = -6, so you would put that under 3 (as shown in the picture). Now, add 3 and -6 (which = -3). Repeat this step each time.

I hope this made sense! Please let me know if you have any questions.

Answer 2

-23

Explanation:

You want to know the remainder from division of 2x³ +3x² -x +1 by (x +3).

Remainder theorem

The remainder theorem tells you the remainder from dividing a polynomial p(x) by (x -q) is the value p(q). Here, this means we can find the remainder by evaluating the polynomial expression for x=-3.

Horner form

It is often convenient to evaluate a polynomial by writing it in Horner form.

2x³ +3x² -x +1 = ((2x +3)x -1)x +1

Then for x = -3, we have ...

((2(-3) +3)(-3) -1)(-3) +1 = (-3(-3) -1)(-3) +1 = 8(-3) +1 = -24 +1 = -23

The remainder from the division is -23.

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Additional comment

If you compare the sum at each stage to the bottom line of a synthetic division tableau, you find they are the same: {-3, 8, -23}.

Horner form uses the minimum number of arithmetic operations to complete the evaluation of the expression: 3 multiplications and 3 additions.

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