Question

The product of two polynomials is $6{x^3} + 29{x^2} + 44x + 21$ and one of the polynomials is $3x + 7$ . Find the other polynomial.

Answer 1

Hint: At first we will determine the degree of the required polynomial by the property that the degree of the product of two polynomials is the sum of degrees of the individuals' polynomials.
After finding the degree of the required polynomial we will assume a polynomial of the same degree, then by substituting that polynomial in the given condition, we will get an equation and on solving and comparing that equation’s expressions we will get the required polynomial.

Complete step-by-step answer:
Given data: The product of two polynomials is $6{x^3} + 29{x^2} + 44x + 21$
One of the polynomials is $3x + 7$
We know that the degree of the product of two polynomials is the sum of degrees of the individuals' polynomials
So from the given data, we can say that the degree of the required polynomial will be 2
Let the polynomial be $a{x^2} + bx + c$
There we can write as, $6{x^3} + 29{x^2} + 44x + 21 = \left( {3x + 7} \right)\left( {a{x^2} + bx + c} \right)$
Solving the right-hand side of the above equation
 $ = \left( {3x + 7} \right)\left( {a{x^2} + bx + c} \right)$
On multiplication and simplification
 $ = 3a{x^3} + 7a{x^2} + 3b{x^2} + 7bx + 3xc + 7c$
Taking the variable constant from the like terms
 $ = 3a{x^3} + \left( {7a + 3b} \right){x^2} + \left( {7b + 3c} \right)x + 7c$
Now comparing it with the right-hand side of the equation i.e. $6{x^3} + 29{x^2} + 44x + 21$
We get, $3a = 6$
Dividing both sides by 3
 $\therefore a = 2$
And $7a + 3b = 29$
Substituting $a = 2$ we get,
 $ \Rightarrow 14 + 3b = 29$
Subtracting both sides by 14 and then dividing by 3 we get,
 $\therefore b = 5$
And $7c = 21$
Dividing both sides by 7 we get,
 $\therefore c = 3$
Therefore the values we get are $a = 2$ , $b = 5$ and $c = 3$
Therefore the required polynomial is $2{x^2} + 5x + 3$.


Note: To get the second polynomial we can use an alternative method i.e.
Let the second polynomial be L
Now we know that $L\left( {3x + 7} \right) = 6{x^3} + 29{x^2} + 44x + 21$
By cross multiplication
 $ \Rightarrow L = \dfrac{{6{x^3} + 29{x^2} + 44x + 21}}{{3x + 7}}$
Therefore dividing the numerator polynomial by denominator polynomial we will get the same answer.