Maths-
General
Easy

Question

If A equals 2 x cubed plus 5 x squared plus 4 x plus 1 text  and  end text B equals 2 x squared plus 3 x plus 1 then find the quotient from the following four option, when A is divided by B.

  1. x − 1
  2. x + 1 
  3. 2x + 1 
  4. 2x - 1 

hintHint:

The expansions of certain identities are:
open parentheses x squared minus a squared close parentheses equals left parenthesis x plus a right parenthesis left parenthesis x minus a right parenthesis
left parenthesis x plus a right parenthesis left parenthesis x plus b right parenthesis equals x squared plus left parenthesis a plus b right parenthesis x plus a b
We are asked to find the quotient from the given options when A is divided by B.

The correct answer is: x + 1


    Step 1 of 2:
    The given expressions are:
    A equals 2 x cubed plus 5 x squared plus 4 x plus 1
    B equals 2 x squared plus 3 x plus 1
    When A is divided by B, we have:
    fraction numerator 2 x cubed plus 5 x squared plus 4 x plus 1 over denominator 2 x squared plus 3 x plus 1 end fraction
    Step 2 of 2:
    Simplify the expression fraction numerator 2 x cubed plus 5 x squared plus 4 x plus 1 over denominator 2 x squared plus 3 x plus 1 end fraction and cancel out the common factors;
    fraction numerator 2 x cubed plus 5 x squared plus 4 x plus 1 over denominator 2 x squared plus 3 x plus 1 end fraction equals fraction numerator open parentheses 2 x cubed plus 4 x squared plus 2 x close parentheses plus open parentheses x squared plus 2 x plus 1 close parentheses plus 2 x over denominator 2 x squared plus 3 x plus 1 end fraction
    equals fraction numerator 2 x open parentheses x squared plus 2 x plus 1 close parentheses plus open parentheses x squared plus 2 x plus 1 close parentheses over denominator 2 x squared plus 2 x plus x plus 1 end fraction
    equals fraction numerator left parenthesis 2 x plus 1 right parenthesis open parentheses x squared plus 2 x plus 1 close parentheses over denominator 2 x left parenthesis x plus 1 right parenthesis plus 1 left parenthesis x plus 1 right parenthesis end fraction
    equals fraction numerator left parenthesis 2 x plus 1 right parenthesis left parenthesis x plus 1 right parenthesis squared over denominator left parenthesis 2 x plus 1 right parenthesis left parenthesis x plus 1 right parenthesis end fraction
    = x +1
    Hence, the quotient is x + 1.
    The correct answer is option b.
    Here, we split the values in the initial steps to group certain terms. This was followed by using the identities to factorize them out.

    Simplify means to make it simple. In mathematics, simplify is the reduction of an expression/fraction into irreducible forms.

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