Question

If the quotient is \[3{x^2} - 2x + 1\], remainder is \[2x - 5\], and the divisor is \[x + 2\], then the dividend is
(a) \[3{x^3} - 4{x^2} + x - 3\]
(b) \[3{x^3} - 4{x^2} - x + 3\]
(c) \[3{x^3} + 4{x^2} - x + 3\]
 (d) \[3{x^3} + 4{x^2} - x - 3\]

Answer 1

Hint: Here, we need to find the dividend. We will use the division algorithm and substitute the values of the quotient, divisor, and the remainder. Then, we will simplify the expression to get the expression for dividend.

Formula Used: The division algorithm states that if \[p\left( x \right)\] and \[g\left( x \right)\] are two polynomials where \[g\left( x \right) \ne 0\], then there are two polynomials \[q\left( x \right)\] and \[r\left( x \right)\] such that \[p\left( x \right) = q\left( x \right) \times g\left( x \right) + r\left( x \right)\]. Here, \[p\left( x \right)\] is the dividend, \[g\left( x \right)\] is the divisor, \[q\left( x \right)\] is the quotient, and \[r\left( x \right)\] is the remainder.

Complete step-by-step answer:
We will use the division algorithm to find the dividend.
The division algorithm states that if \[p\left( x \right)\] and \[g\left( x \right)\] are two polynomials where \[g\left( x \right) \ne 0\], then there are two polynomials \[q\left( x \right)\] and \[r\left( x \right)\] such that \[p\left( x \right) = q\left( x \right) \times g\left( x \right) + r\left( x \right)\]. Here, \[p\left( x \right)\] is the dividend, \[g\left( x \right)\] is the divisor, \[q\left( x \right)\] is the quotient, and \[r\left( x \right)\] is the remainder.
It is given that the quotient is \[3{x^2} - 2x + 1\], remainder is \[2x - 5\], and the divisor is \[x + 2\].
Thus, we get
\[q\left( x \right) = 3{x^2} - 2x + 1\]
\[r\left( x \right) = 2x - 5\]
\[g\left( x \right) = x + 2\]
We need to find the dividend, that is \[p\left( x \right)\].
Substituting \[q\left( x \right) = 3{x^2} - 2x + 1\], \[r\left( x \right) = 2x - 5\], and \[g\left( x \right) = x + 2\] in the division algorithm, we get
\[ \Rightarrow p\left( x \right) = \left( {3{x^2} - 2x + 1} \right)\left( {x + 2} \right) + \left( {2x - 5} \right)\]
Multiplying the terms using the distributive law of multiplication, we get
\[ \Rightarrow p\left( x \right) = 3{x^3} + 6{x^2} - 2{x^2} - 4x + x + 2 + 2x - 5\]
Adding and subtracting the like terms, we get
\[ \Rightarrow p\left( x \right) = 3{x^3} + 4{x^2} - x - 3\]
Therefore, we get the dividend as \[3{x^3} + 4{x^2} - x - 3\].
Thus, the correct option is option (d).

Note: We have used the distributive law of multiplication in the solution to verify our answer. The distributive law of multiplication states that \[\left( {a + b + c} \right)\left( {d + e} \right) = a \cdot d + a \cdot e + b \cdot d + b \cdot e + c \cdot d + c \cdot e\].
We added and subtracted the like terms in the expression \[3{x^3} + 6{x^2} - 2{x^2} - 4x + x + 2 + 2x - 5\]. Like terms are the terms whose variables and their exponents are the same. For example, \[100x,150x,240x,600x\] all have the variable \[x\] raised to the exponent 1. Terms which are not like, and have different variables, or different degrees of variables cannot be added together. For example, we can add the term \[x\] to \[2x\], but we cannot add \[x\] to \[3{x^3}\], or \[6{x^2}\], or 2.