To find our quotient we are going to perform long division.
Long division step by step:
Step 1. Write the polynomial (dividend) in descending order (from highest to least exponent). If you encounter a missing term, use zero to fill the space of the term in the descending sequence.
Notice that even tough our polynomial is written in descending form, [tex] x^3 [/tex] is missing, so we are going to use [tex] 0x^3 [/tex] to fill the gap:
[tex] 3x^4-4x^2+8x-1 [/tex]
[tex] 3x^4+0x^3-4x^2+8x-1 [/tex]
Step 2. Divide the first term of the polynomial (dividend) by the first term of the divisor.
In our case the first term of the dividend is [tex] 3x^4 [/tex] and the first term of the divisor is [tex] x [/tex], so [tex] \frac{3x^4}{x} =3x^3 [/tex]. Notice that [tex] 3x^3 [/tex] is the first term of the quotient.
Step 3. Multiply the first term of the quotient by the terms of the divisor and subtract them from the respective term of the dividend.
The first term of our quotient (from the previous step) is [tex] 3x^3 [/tex] and the divisor is [tex] (x-1) [/tex], so [tex] 3x^3(x-2)=3x^4-6x^3 [/tex]. Now, we are going to subtract them from the respective term (the term with the same power) of the dividend. The respective terms of the dividend are [tex] 3x^4 [/tex] and [tex] 0x^3 [/tex], so [tex] 3x^4-3x^4=0 [/tex] and [tex] 0x^3-(-6x^3)=0x^3+6x^3=6x^3 [/tex]
Step 4. Bring down the next term in the dividend and repeat the process for the remaining terms.
After finish the process (check the attached picture), we can conclude that the quotient of [tex] 3x^4-4x^2+8x-1 [/tex] ÷ [tex] (x-2) [/tex] is [tex] 3x^3+6x^2+8x+24 [/tex] with a remainder of [tex] 47 [/tex], or in a different notation: [tex] 3x^3+6x^2+8x+24+\frac{47}{x-2} [/tex]
