hi guys!

I was wondering how to do x6-18x2+8 divided by factor x2-4

I normally use synethetic division what else easier method should i use?

since x^2 - 4 = (x+2)(x-2) use synthetic division twice, it works out

make sure you allow for the missing terms in the line-up

1 0 0 0-18 0 8

I ended up with x^4 + 4x^2 - 2

thank you so much!

also for x3+1 what is the factor??

is it isquareroot1

Hi! To divide x^6 - 18x^2 + 8 by the factor x^2 - 4, you can indeed use synthetic division, which is a common and efficient method. However, I can also show you an alternative approach using polynomial long division. Let's go through the steps together:

Step 1: Write the dividend (x^6 - 18x^2 + 8) and the divisor (x^2 - 4) in descending order of powers of x.

x^6 - 18x^2 + 8
÷ x^2 - 4

Step 2: Divide the highest-degree term of the dividend (x^6) by the highest-degree term of the divisor (x^2). The result is the first term of the quotient:

x^6 ÷ x^2 = x^4

Step 3: Multiply the divisor (x^2 - 4) by the term just found (x^4) and write the product below the dividend, aligned according to their corresponding powers of x:

x^4
__________
x^2 - 4 │ x^6 - 18x^2 + 8

x^6 - 4x^4

Step 4: Subtract the product obtained in step 3 from the dividend:

x^6 - 18x^2 + 8 - (x^6 - 4x^4) = -4x^4 - 18x^2 + 8

Step 5: Bring down the next term of the dividend (-4x^4):

x^4 - 4x^4

___________
x^2 - 4 │ x^6 - 18x^2 + 8
x^6 - 4x^4
____________
-14x^4 - 18x^2 + 8

Step 6: Divide the highest-degree term of the new dividend (-14x^4) by the highest-degree term of the divisor (x^2):

-14x^4 ÷ x^2 = -14x^2

Step 7: Multiply the divisor (x^2 - 4) by the term just found (-14x^2) and write the product below the new dividend:

x^4 - 4x^4 - 14x^2

Step 8: Subtract the product obtained in step 7 from the new dividend:

-14x^4 - 18x^2 + 8 - (x^4 - 4x^4 - 14x^2) = -18x^2 - 14x^4 + 8x^2 + 8

Step 9: Bring down the next term of the dividend (-18x^2):

x^4 - 4x^4 - 14x^2 + 8x^2

Step 10: Repeat steps 6-9 until all terms have been processed.

Finally, you will obtain the quotient of x^4 - 14x^2 - 4 and a remainder of 8x^2 + 8. Therefore, the division of x^6 - 18x^2 + 8 by the factor x^2 - 4 is equal to x^4 - 14x^2 - 4, with a remainder of 8x^2 + 8.

Note: Polynomial long division can be a bit more time-consuming than synthetic division, especially for higher-degree polynomials. However, it provides a step-by-step process that helps you understand the concept more thoroughly.