How do you divide #(4x^5 +8x^4 -8x^3 +4x^2 +x-8) / (5x^2 -4x+9)#?

Question

Abigail Allen · Accepted Answer

Since coefficient of highest power of #x#, i.e ., #x^2# term in the denominator is #!=1#, therefore we need to divide using the long division method.
Quotient #=4/5x^3+56/25x^2-156/125x+1396/625#
Remainder#=13229/625x-17564/625# #color(white)(WWWWWWW)4/5x^3+56/25x^2-156/125x+1396/625# #5x^2-4x+9)bar(4x^5+8x^4-8x^3+4x^2+x-8)(# #color(white)(WWWWWW)4x^5-16/5x^4+36/5x^3# #color(white)(WWWWW)ul(-color(white)(iiW)+color(white)(WiW)-color(white)(WWWWW))# #color(white)(WWWWWWWWW)56/5x^4-76/5x^3+" "4x^2# #color(white)(WWWWWWWWW)56/5x^4-224/25x^3+504/25x^2# #color(white)(WWWWWWWW)ul(-color(white)(WiiW)+color(white)(WiW)-color(white)(WWW))# #color(white)(WWWWWWWWWWW)-156/25x^3+404/25x^2+x# #color(white)(WWWWWWWWWWW)-156/25x^3+624/125x^2-1404/125x# #color(white)(WWWWWWIWW)ul(color(white)(WiiW)+color(white)(WWW)-color(white)(iWW)+)# #color(white)(WWWWWWWWWWWWW)1396/125x^2+1529/125x-8# #color(white)(WWWWWWWWWWWWW)1396/125x^2-5584/625x+12564/625# #color(white)(WWWWWWWiWW)ul(color(white)(WiiW)-color(white)(WWW)+color(white)(WiWW)-)# #color(white)(WWWWWWWWWWWWWWWWW)13229/625x-17564/625#

Isaiah Anthony · Answer

undefined To divide (4x^5 + 8x^4 - 8x^3 + 4x^2 + x - 8) by (5x^2 - 4x + 9), you can use long division. Here are the steps:

1. Start by dividing the highest degree term of the numerator (4x^5) by the highest degree term of the denominator (5x^2). The result is (4/5)x^(5-2) = (4/5)x^3.

2. Multiply the entire denominator (5x^2 - 4x + 9) by the result obtained in step 1, which is (4/5)x^3. This gives you (4/5)x^3 * (5x^2 - 4x + 9) = (4/5)x^3 * 5x^2 - (4/5)x^3 * 4x + (4/5)x^3 * 9.

3. Subtract the result obtained in step 2 from the numerator (4x^5 + 8x^4 - 8x^3 + 4x^2 + x - 8). This gives you (4x^5 + 8x^4 - 8x^3 + 4x^2 + x - 8) - [(4/5)x^3 * 5x^2 - (4/5)x^3 * 4x + (4/5)x^3 * 9].

4. Simplify the expression obtained in step 3 by combining like terms.

5. Repeat steps 1-4 with the simplified expression obtained in step 4 until you have no more terms to divide.

6. The final result will be the quotient obtained from the division.

Note: It is important to carefully perform each step and simplify the expression after each subtraction to ensure accuracy in the division process.