How do you divide #( -3x^3+ 4x^2 + 12x + 9 )/(x - 2 )#?

Question

How do you divide #( -3x^3+ 4x^2 + 12x + 9 )/(x - 2  )#?

Eli Assouline · Accepted Answer

#(-3x^3+4x^2+12x+9)/(x-2) " "=" "-3x^2-2x+8+25/(x-2)#

#" "-3x^3+4x^2+12x+9" ............checked"#
#color(red)(-3x^2)(x-2)-> ul(-3x^3+6x^2) larr" subtract .........corrected"#
#" "0-2x^2+12x+9#
#color(red)(-2x)(x-2)->" "ul(-2x^2+4x)larr" subtract"#
#" "0+8x+9#
#color(red)(+8)(x-2)->" "ul(8x-16)larr" subtract"#
#" "0color(red)(+25 larr" remainder")#

#(-3x^3+4x^2+12x+9)/(x-2) " "=" "color(red)(-3x^2-2x+8+)color(red)(25)/(x-2)#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Checked in Maple:

Ethan Adkins · Answer

undefined To divide (-3x^3 + 4x^2 + 12x + 9) by (x - 2), you can use long division or synthetic division. Here is the step-by-step process using long division:

1. Divide the first term of the dividend (-3x^3) by the first term of the divisor (x). This gives you -3x^2.
2. Multiply the divisor (x - 2) by the quotient obtained in step 1 (-3x^2). This gives you (-3x^2)(x - 2) = -3x^3 + 6x^2.
3. Subtract the result obtained in step 2 from the original dividend (-3x^3 + 4x^2 + 12x + 9) to get the new dividend: (4x^2 + 12x + 9) - (-3x^3 + 6x^2) = -3x^3 + 4x^2 + 12x + 9 + 3x^3 - 6x^2 = 10x^2 + 12x + 9.
4. Repeat steps 1-3 with the new dividend (10x^2 + 12x + 9) until the degree of the new dividend is less than the degree of the divisor.

Continuing the process:
5. Divide the first term of the new dividend (10x^2) by the first term of the divisor (x). This gives you 10x.
6. Multiply the divisor (x - 2) by the quotient obtained in step 5 (10x). This gives you (10x)(x - 2) = 10x^2 - 20x.
7. Subtract the result obtained in step 6 from the new dividend (10x^2 + 12x + 9) to get the new dividend: (12x + 9) - (10x^2 - 20x) = 10x^2 - 20x + 12x + 9 = 10x^2 - 8x + 9.
8. Repeat steps 5-7 with the new dividend (10x^2 - 8x + 9) until the degree of the new dividend is less than the degree of the divisor.

Continuing the process:
9. Divide the first term of the new dividend (10x^2) by the first term of the divisor (x). This gives you 10x.
10. Multiply the divisor (x - 2) by the quotient obtained in step 9 (10x). This gives you (10x)(x - 2) = 10x^2 - 20x.
11. Subtract the result obtained in step 10 from the new dividend (10x^2 - 8x + 9) to get the remainder: (-8x + 9) - (10x^2 - 20x) = -10x^2 + 12x + 9.

Therefore, the quotient is -3x^2 + 10x + 10, and the remainder is -10x^2 + 12x + 9.