Question

How do you divide \[\left( {{x^3} + 15{x^2} + 45x - 25} \right) \div \left( {x + 5} \right)\] using synthetic division?

Answer 1

Hint:
Here we will use the basics of division operation. First, we will factor out common terms from the given dividend until we get one of the factors in such a form that it will have \[x + 5\]. Then we will cancel out the common term to get the required answer of the division.

Complete step by step solution:
Dividend given is \[\left( {{x^3} + 15{x^2} + 45x - 25} \right)\] and divisor is \[\left( {x + 5} \right)\].
\[\dfrac{{{x^3} + 15{x^2} + 45x - 25}}{{x + 5}}\]
Now we will divide the dividend by the divisor to get the quotient. So, we will modify the dividend of the equation such that it will have \[x + 5\]. Therefore the equation can be written as
\[ \Rightarrow \dfrac{{{x^3} + 15{x^2} + 45x - 25}}{{x + 5}} = \dfrac{{{x^3} + 5{x^2} + 10{x^2} + 50x - 5x - 25}}{{x + 5}}\]
Now we will take the commons from the terms in the numerator. Therefore, we get
\[ \Rightarrow \dfrac{{{x^3} + 15{x^2} + 45x - 25}}{{x + 5}} = \dfrac{{{x^2}\left( {x + 5} \right) + 10x\left( {x + 5} \right) - 5\left( {x + 5} \right)}}{{x + 5}}\]
Now we will take \[x + 5\] common in the numerator of the equation. Therefore, we get
\[ \Rightarrow \dfrac{{{x^3} + 15{x^2} + 45x - 25}}{{x + 5}} = \dfrac{{\left( {x + 5} \right)\left( {{x^2} + 10x - 5} \right)}}{{x + 5}}\]
Now we will cancel out the common terms in the numerator and denominator. Therefore, we get
\[ \Rightarrow \dfrac{{\left( {x + 5} \right)\left( {{x^2} + 10x - 5} \right)}}{{x + 5}} = {x^2} + 10x - 5\]
Therefore, we can write the above division as
\[ \Rightarrow {x^3} + 15{x^2} + 45x - 25 = \left[ {\left( {x + 5} \right) \times \left( {{x^2} + 10x - 5} \right)} \right] + 0\]

Hence, the given division i.e. \[\left( {{x^3} + 15{x^2} + 45x - 25} \right) \div \left( {x + 5} \right)\] using synthetic division is equal to \[{x^2} + 10x - 5\]. So, the quotient of the equation is \[{x^2} + 10x - 5\] and the remainder of the equation is 0.

Note:
We should know that the division is the operation in which the dividend is divided by the divisor to get the quotient along with some remainder. So, the general formula of the division operation is
\[{\text{Dividend}} = \left( {{\text{Divisor}} \times {\text{Quotient}}} \right) + {\text{Remainder}}\]
The dividend is the term or number which is to be divided. The divisor is the term or number which is divided. The quotient is the term or number which is the answer to this division operation and the remainder is the term which is left when a division operation is performed.