Algebra 2 plz help me dived

Question

. Divide: (12𝑥3−11𝑥2+9𝑥+18)÷(4𝑥+3)

Answer 1

Dear Caleb

Let us assume that we do not know anything about long division or synthetic division.

When the polynomial (12𝑥3−11𝑥2+9𝑥+18) is divided by the binomial (4𝑥+3)

we expect as quotient a trinomial say a x² + b x + c and as a remainder a number say k.

Then (12𝑥3−11𝑥2+9𝑥+18) =(4𝑥+3)( a x² + b x + c )+ k, then performing the multiplication on the right side

of the equation and collecting like terms we have

12𝑥3−11𝑥2+9𝑥+18 = 4ax³ + ( 4b + 3a ) x² + (4c + 3b) x +3c +k

Then equating the corresponding coefficients we obtain

12 = 4a ⇒ a=3

-11 = 4b + 3a and since a = 3 then -20 = 4b ⇒ b = -5

9 = 4c + 3b and since b = -5 then 24 = 4 c ⇒c = 6

18 = 3c +k and since c = 6 then k = 0

There fore the quotient is a x² + b x + c = 3x^2 -5x + 6 and the remainder is equal to zero.

Of course for this kind of problem long division or synthetic division are the preferable methods,

but i wanted to point out to you that always there is another way...

Answer 2

3x^2 -5x + 6 is the answer