How do you divide #(6x^3 + 10x^2 + x + 8) / (2x^2 + 1)#?

Question

Isaiah Anthony · Accepted Answer

#(6x^3+10x^2+x+8)/(2x^2+1) = 3x+5# with remainder #(-2x+3)#

Using polynomial long division: #{: (,,3x,+5,,), (,,"----","----","----","----"), (2x^2+1,")",6x^3,+10x^2,+x,+8), (,,6x^3,,+3x,), (,,"----","----","----","----"), (,,,10x^2,-2x,+8), (,,,10x^2,,+5), (,,,"----","----","----"), (,,,,-2x,+3) :}#

Avery Alexander · Answer

undefined To divide (6x^3 + 10x^2 + x + 8) by (2x^2 + 1), you can use long division.

First, divide the highest degree term of the numerator (6x^3) by the highest degree term of the denominator (2x^2). This gives you 3x as the first term of the quotient.

Next, multiply the entire denominator (2x^2 + 1) by the first term of the quotient (3x), and subtract the result from the numerator (6x^3 + 10x^2 + x + 8). This gives you (6x^3 + 10x^2 + x + 8) - (6x^3 + 3x) = (10x^2 - 2x + 8).

Now, repeat the process with the new numerator (10x^2 - 2x + 8) and the original denominator (2x^2 + 1). Divide the highest degree term (10x^2) by the highest degree term (2x^2) to get 5 as the next term of the quotient.

Multiply the entire denominator (2x^2 + 1) by the new term of the quotient (5), and subtract the result from the new numerator (10x^2 - 2x + 8). This gives you (10x^2 - 2x + 8) - (10x^2 + 5) = (-2x + 3).

Repeat the process with the new numerator (-2x + 3) and the original denominator (2x^2 + 1). Divide the highest degree term (-2x) by the highest degree term (2x^2) to get -x/2 as the next term of the quotient.

Multiply the entire denominator (2x^2 + 1) by the new term of the quotient (-x/2), and subtract the result from the new numerator (-2x + 3). This gives you (-2x + 3) - (-x) = (x + 3).

Since the degree of the new numerator (x + 3) is less than the degree of the denominator (2x^2 + 1), you have reached the end of the division.

Therefore, the quotient is 3x + 5 - (x/2) and the remainder is (x + 3).