How do you divide #(2x^4 -x^3 - 9x^2 - x + 5)/ (5x^2 - 2)
# using polynomial long division?

Question

How do you divide #(2x^4 -x^3 - 9x^2 - x + 5)/ (5x^2 - 2)  
# using polynomial long division?

Abigail Allen · Accepted Answer

This really is long division but its layout is different to the conventional approach.

#2/5x^2-1/5x-41/25+(-35x+44)/(25 (5x^2-2))# Given: #(color(brown)(2x^4-x^3-9x^2-x+5))/(color(green)(5x^2-2))# Using place keepers of no value. Example: #0x^3# #color(white)("ddddddddddddddd")color(brown)(2x^4-color(white)("d")x^3color(white)("d")-9x^2-x+5)# #color(magenta)(+2/5x^2)color(green)((5x^2-2)) ->ul(2x^4+0x^3-4/5x^2larr" Subtract")# #color(white)("ddddddddddddddddd")0-x^3-41/5x^2-x+5 # #color(magenta)(-1/5)color(green)((5x^2-2)) ->color(white)("d.dd")ul( -x^3+color(white)("d.")0x^2+2/5xlarr" Subtract")# #color(white)("ddddddddddddddddddddd")0-41/5x^2-7/5x+5 # #color(magenta)(-41/25)color(green)((5x^2-2)) ->color(white)("dddddddd")ul(-41/5x^2+0x+81/25larr" Sub.")# #color(white)("d")color(magenta)("Remainder "->color(white)("ddddddddddddd")0-7/5x+44/25)# #color(magenta)("Set remainder as: "(-35x+44)/25)# giving: #color(magenta)(2/5x^2-1/5x-41/25+[(-35x+44)/25 color(green)(-:(5x^2-2))])# #color(magenta)(2/5x^2-1/5x-41/25+[(-35x+44)/(25 color(green)((5x^2-2)))])#

Savannah Anderson · Answer

The remainder is #=(-7/5x+43/25)# and the quotient is #=(2/5x^2-x/5-41/25)#

Apply the polynomial long division formula.

#color(white)(aaaa)##2x^4-x^3-9x^2-x+5##color(white)(aaaa)##|##5x^2-2#

#color(white)(aaaaaaaaaaaaaaaaaaaaaaaa)##color(white)(aaaa)##|##2/5x^2-x/5-41/25#

#color(white)(aaaa)##2x^4-0x^3-4/5x^2#

#color(white)(aaaa)##0x^4-1x^3-41/5x^2-x#

#color(white)(aaaaaaaa)##-1x^3-00x^2+2/5x#

#color(white)(aaaaaaaa)##-0x^3-41/5x^2-7/5x+5#

#color(white)(aaaaaaaaaaaaa)##-41/5x^2-7/5x+82/25#

#color(white)(aaaaaaaaaaaaaaa)##-0x^2-7/5x+43/25#

Consequently,

#(2x^4-x^3-9x^2-x+5)/(5x^2-2)=(2/5x^2-x/5-41/25)+(-7/5x+43/25)/(5x^2-2)#

The remainder is #=(-7/5x+43/25)# and the quotient is #=(2/5x^2-x/5-41/25)#

Avery Alexander · Answer

undefined To divide (2x^4 - x^3 - 9x^2 - x + 5) by (5x^2 - 2) using polynomial long division, follow these steps:

1. Arrange the dividend and divisor in descending order of exponents.
   Dividend: 2x^4 - x^3 - 9x^2 - x + 5
   Divisor: 5x^2 - 2

2. Divide the leading term of the dividend (2x^4) by the leading term of the divisor (5x^2), which gives (2x^4)/(5x^2) = (2/5)x^2.

3. Multiply the divisor (5x^2 - 2) by the result obtained in step 2, (2/5)x^2. This gives (2/5)x^2 * (5x^2 - 2) = (2/5)x^4 - (4/5)x^2.

4. Subtract the result obtained in step 3 from the dividend. (2x^4 - x^3 - 9x^2 - x + 5) - ((2/5)x^4 - (4/5)x^2) = (8/5)x^4 - x^3 - (1/5)x^2 - x + 5.

5. Bring down the next term from the dividend, which is -x. The new dividend becomes (8/5)x^4 - x^3 - (1/5)x^2 - x + 5 - x = (8/5)x^4 - x^3 - (1/5)x^2 - 2x + 5.

6. Repeat steps 2-5 until the degree of the new dividend is less than the degree of the divisor.

7. Divide the leading term of the new dividend ((8/5)x^4) by the leading term of the divisor (5x^2), which gives ((8/5)x^4)/(5x^2) = (8/25)x^2.

8. Multiply the divisor (5x^2 - 2) by the result obtained in step 7, (8/25)x^2. This gives (8/25)x^2 * (5x^2 - 2) = (8/25)x^4 - (16/25)x^2.

9. Subtract the result obtained in step 8 from the new dividend. (8/5)x^4 - x^3 - (1/5)x^2 - 2x + 5 - ((8/25)x^4 - (16/25)x^2) = (32/25)x^4 - x^3 + (9/25)x^2 - 2x + 5.

10. Bring down the next term from the new dividend, which is -2x. The new dividend becomes (32/25)x^4 - x^3 + (9/25)x^2 - 2x + 5 - (-2x) = (32/25)x^4 - x^3 + (9/25)x^2.

11. Repeat steps 7-10 until the degree of the new dividend is less than the degree of the divisor.

12. Divide the leading term of the new dividend ((32/25)x^4) by the leading term of the divisor (5x^2), which gives ((32/25)x^4)/(5x^2) = (32/125)x^2.

13. Multiply the divisor (5x^2 - 2) by the result obtained in step 12, (32/125)x^2. This gives (32/125)x^2 * (5x^2 - 2) = (32/125)x^4 - (64/125)x^2.

14. Subtract the result obtained in step 13 from the new dividend. (32/25)x^4 - x^3 + (9/25)x^2 - ((32/125)x^4 - (64/125)x^2) = (96/125)x^4 - x^3 + (73/125)x^2.

15. The degree of the new dividend (96/125)x^4 - x^3 + (73/125)x^2 is less than the degree of the divisor (5x^2 - 2), so the division is complete.

The quotient is (2/5)x^2 + (8/25)x^2 + (32/125)x^2 = (2/5)x^2 + (8/25)x^2 + (32/125)x^2 = (2/5)x^2 + (8/25)x^2 + (32/125)x^2.

The remainder is (96/125)x^4 - x^3 + (73/125)x^2.

Jace Anderson · Answer

undefined To divide (2x^4 - x^3 - 9x^2 - x + 5) by (5x^2 - 2) using polynomial long division:

1. Divide the leading term of the dividend (2x^4) by the leading term of the divisor (5x^2) to get (2x^4) / (5x^2) = (2/5)x^2.
2. Multiply the entire divisor by (2/5)x^2 to get (2/5)x^4 - (4/5)x^2.
3. Subtract this result from the dividend to get the remainder: (2x^4 - x^3 - 9x^2 - x + 5) - ((2/5)x^4 - (4/5)x^2) = (8/5)x^4 - x^3 - (1/5)x^2 - x + 5.
4. Repeat the process with the new polynomial (8/5)x^4 - x^3 - (1/5)x^2 - x + 5.

Continue this process until the degree of the remainder is less than the degree of the divisor.

How do you divide #(2x^4 -x^3 - 9x^2 - x + 5)/ (5x^2 - 2) # using polynomial long division?