How do you divide #(x^4 + 5x^2 + 12x + 3)/( 6x+8)#?

Question

How do you divide #(x^4 +  5x^2 + 12x + 3)/( 6x+8)#?

Hazel Anglin · Accepted Answer

#=> x^3/6-(2x^2)/9+(61x)/54+40/81-77/(486x+648)# #" "x^4+0x^3+5x^2+12x+3# #color(magenta)(1/6 x^3)(6x+8) -> " "color(white)(a)ul(x^4+4/3 x^3" ")" "larr "Subtract"# #" "0 - 4/3x^3 +5x^2+12x+3# #color(magenta)(-2/9x^2)(6x+8) ->" "color(white)(a)ul(-4/3x^3-16/9x^2" ")" "larr "Subtract"# #" "0 + 61/9x^2+12x+3# #color(magenta)(61/54x)(6x+8) ->" "color(white)(a)ul(61/9x^2+244/27x)" "larr "Subtract"# #" "0+80/27x+3# #color(magenta)(40/81)(6x+8) =>" "color(white)(a)ul(80/27x+320/81)" Subtract"# #" "0-77/81# #=> x^3/6-(2x^2)/9+(61x)/54+40/81-77/(81(6x+8)# #=> x^3/6-(2x^2)/9+(61x)/54+40/81-77/(486x+648)#

Eva Adamson · Answer

undefined To divide (x^4 + 5x^2 + 12x + 3) by (6x + 8), you can use long division. Here are the steps:

1. Divide the first term of the numerator (x^4) by the first term of the denominator (6x). This gives you (1/6)x^3.

2. Multiply the entire denominator (6x + 8) by (1/6)x^3, and subtract the result from the numerator. This gives you a new numerator: (-2/3)x^3 + 5x^2 + 12x + 3.

3. Bring down the next term from the numerator, which is 5x^2.

4. Divide the first term of the new numerator (-2/3)x^3 by the first term of the denominator (6x). This gives you (-1/9)x^2.

5. Multiply the entire denominator (6x + 8) by (-1/9)x^2, and subtract the result from the new numerator. This gives you a new numerator: (1/3)x^2 + 12x + 3.

6. Bring down the next term from the numerator, which is 12x.

7. Divide the first term of the new numerator (1/3)x^2 by the first term of the denominator (6x). This gives you (1/18)x.

8. Multiply the entire denominator (6x + 8) by (1/18)x, and subtract the result from the new numerator. This gives you a new numerator: (1/3)x + 3.

9. Bring down the next term from the numerator, which is 3.

10. Divide the first term of the new numerator (1/3)x by the first term of the denominator (6x). This gives you (1/18).

11. Multiply the entire denominator (6x + 8) by (1/18), and subtract the result from the new numerator. This gives you a new numerator: 3.

12. Since the new numerator (3) is a constant term, you have reached the end of the division.

Therefore, the result of dividing (x^4 + 5x^2 + 12x + 3) by (6x + 8) is (1/6)x^3 - (1/9)x^2 + (1/18)x + (3/(6x + 8)).