Question

A polynomial $p(x)$ is divided by $2x - 1$. The quotient and remainder obtained are $(7{x^2} + x + 5)$ and $4$ respectively. Find $p(x)$.

Answer 1

Hint:
Use the quotient remainder theorem for the polynomials which states that a polynomial is equal to the product of the divisor and quotient plus remainder.
Mathematically we can express this as,
$p(x) = d(x)q(x) + r(x)$

Complete step-by-step answer:
We are given that a polynomial $p(x)$ is divided by $2x - 1$. The quotient and remainder obtained are $(7{x^2} + x + 5)$ and $4$ respectively.
We have to find the value of the polynomial $p(x)$.
Since the polynomial is divided by $2x - 1$ therefore, the value of the divisor $d(x)$ is $2x - 1$.
The quotient and remainder are $(7{x^2} + x + 5)$ and $4$ respectively.
Therefore, $q(x) = 7{x^2} + x + 5$ and $r(x) = 4$
Now, we use the quotient remainder theorem for the polynomials.
The theorem states that a polynomial is equal to the product of the divisor and quotient plus remainder.
It means $p(x) = d(x)q(x) + r(x)$
Substitute the value of $d(x)$, $q(x)$and $r(x)$.
$\therefore p(x) = (7{x^2} + x + 5)(2x - 1) + 4$
Use distributive property of multiplication to evaluate the product of the polynomials.
The property as follows;
$a(x + y) = ax + ay$
Solve the equation and evaluate the value of $p(x)$.
$ \Rightarrow p(x) = 14{x^3} + 2{x^2} + 10x - 7{x^2} - x - 5 + 4$
Combine the like terms and simplify the expression.
$
   \Rightarrow p(x) = 14{x^3} + 2{x^2} - 7{x^2} + 10x - x - 5 + 4 \\
   \Rightarrow p(x) = 14{x^3} - 5{x^2} + 9x - 1 \\
 $
Therefore, the value of the polynomial $p(x)$ is $14{x^3} - 5{x^2} + 9x - 1$.

Note:
While multiplying the polynomials using the distributive law don’t forget to check the signs of the terms. If the signs are placed wrongly, the answer would be different.
If the value of the remainder is zero in the quotient remainder theorem for the polynomials, then we say that the divisor and the quotient are the factors of the polynomial.
Like terms in the polynomial are the terms which have the same variables.