Question

What is the quotient: (6x4 + 15x3 + 10x2 + 10x + 4) ÷ (3x2 + 2)?

a. 2x2 – 5x + 2
b. 2x2 + 5x – 2
c. 2x2 + 5x + 2
d. 2x2 – 5x – 2

Answer 1

2x^2 + 5x + 2
3x^2 + 2 6x^4 + 15x^3 + 10x^2 + 10x + 4
- 6x^4 + 4x^2
15x^3 + 6x^2 + 10x + 4
- 15x^3 + 10x
6x^2 + 4
6x^2 + 4

Therefore, (6x^4 + 15x^3 + 10x^2 + 10x + 4) ÷ (3x^2 + 2) = 2x^2 + 5x + 2

Answer 2

Answer: Option 'C' is correct.

Explanation:

Since we have given that

`(\left(6x^4+15x^3+10x^2+10x+4\right))/(\left(3x^2+2\right))`

Now, we will find the quotient by factoring the numerator:

`\mathrm{Use\:the\:rational\:root\:theorem}\\a_0=4,\:\quad a_n=6\\\\\mathrm{The\:dividers\:of\:}a_0:\quad 1,\:2,\:4,\:\quad \\\mathrm{The\:dividers\:of\:}a_n:\quad 1,\:2,\:3,\:6\\\\\mathrm{Therefore,\:check\:the\:following\:rational\:numbers:\quad }\pm (1,\:2,\:4)/(1,\:2,\:3,\:6)\\\\-(2)/(1)\mathrm{\:is\:a\:root\:of\:the\:expression,\:so\:factor\:out\:}x+2\\\\=\left(x+2\right)(6x^4+15x^3+10x^2+10x+4)/(x+2)\\\\=(6x^4+15x^3+10x^2+10x+4)/(x+2)=6x^3+3x^2+4x+2\\\\`

Now, we will factor it again:

`=\left(6x^3+3x^2\right)+\left(4x+2\right)\\\\=2\left(2x+1\right)+3x^2\left(2x+1\right)\\\\=\left(2x+1\right)\left(3x^2+2\right)`

At last we get our factorised form :

`=\left(x+2\right)\left(2x+1\right)\left(3x^2+2\right)\\\\=(\left(x+2\right)\left(2x+1\right)\left(3x^2+2\right))/(3x^2+2)\\\\=\left(x+2\right)\left(2x+1\right)\\\\=2x^2+5x+2`

Hence, Option 'C' is correct.