How do find the quotient of #(2x^3 − 3x 2 + x − 6) ÷ (x − 4)#?

Question

How do find the quotient of #(2x^3 − 3x^2 + x − 6) ÷ (x − 4)#?

Savannah Anderson · Accepted Answer

The quotient polynomial :
#q(x)=2x^2+5x+21 and"the Remainder"=78# #(2x^3-3x^2+x-6)div(x-4)# Using synthetic division : We have , #p(x)=(2x^3-3x^2+x-6) and "divisor : " x=4# We take ,coefficients of #p(x) to 2,-3,1,-6# . #4 |# #2color(white)(....)-3color(white)(.......)1color(white)(.......)-6# #ulcolor(white)(...)|# #ul(0color(white)( .........)8color(white)(.......)20color(white)(.........)84# #color(white)(....)2color(white)(........)5color(white)(.......)21color(white)(......)color(violet)(ul|78|# We can see that , quotient polynomial : #q(x)=2x^2+5x+21 and"the Remainder"=78# Hence , #(2x^3-3x^2+x-6)=(x-4)(2x^2+5x+21 )+(78)#

Eli Assouline · Answer

undefined To find the quotient of (2x^3 − 3x^2 + x − 6) ÷ (x − 4), you can use long division. Here are the steps:

1. Divide the first term of the dividend (2x^3) by the first term of the divisor (x). The result is 2x^2.
2. Multiply the divisor (x − 4) by the quotient obtained in step 1 (2x^2). The result is 2x^3 − 8x^2.
3. Subtract the result obtained in step 2 from the dividend (2x^3 − 3x^2 + x − 6) to get the new dividend: (-8x^2 + x − 6).
4. Repeat steps 1-3 with the new dividend (-8x^2 + x − 6).
5. Divide the first term of the new dividend (-8x^2) by the first term of the divisor (x). The result is -8x.
6. Multiply the divisor (x − 4) by the quotient obtained in step 5 (-8x). The result is -8x^2 + 32x.
7. Subtract the result obtained in step 6 from the new dividend (-8x^2 + x − 6) to get the new dividend: (33x − 6).
8. Repeat steps 5-7 with the new dividend (33x − 6).
9. Divide the first term of the new dividend (33x) by the first term of the divisor (x). The result is 33.
10. Multiply the divisor (x − 4) by the quotient obtained in step 9 (33). The result is 33x − 132.
11. Subtract the result obtained in step 10 from the new dividend (33x − 6) to get the remainder: (126).
12. The quotient is the sum of the quotients obtained in each step: 2x^2 - 8x + 33.