Question

Find the HCF of $10{x^2} - x - 21$ and $8{x^3} - 14{x^2} - x + 6$
A) $2x - 3$
B) $2x + 3$
C) $5x + 7$
D) $5x - 7$

Answer 1

Hint: The given question deals with the concept of HCF. HCF stands for the highest common factor. It is the highest number which can be divided into exactly two or more numbers without leaving any remainders. It is generally known as the greatest common divisor or GCD. There are three ways to find out the greatest common factor of any given two or more numbers:
-Factorization method
-Prime Factorization method
-Division method

Complete step by step solution:
Given equations are $10{x^2} - x - 21$ and $8{x^3} - 14{x^2} - x + 6$
We have to find the highest common factor.
Let us simplify the equation $10{x^2} - x - 21$ using the method of splitting the middle term, we get,
$
   \Rightarrow 10{x^2} - x - 21 \\
   \Rightarrow 10{x^2} - 15x + 14x - 21 \\
   \Rightarrow 5x\left( {2x - 3} \right) + 7\left( {2x - 3} \right) \\
   \Rightarrow \left( {2x - 3} \right)\left( {5x + 7} \right) \\
 $
From here we get the value of $x$ as $\dfrac{3}{2}$ and $ - \dfrac{7}{5}$.
Now we will substitute both the values of $x$ one by one and see which one of them divides equation $8{x^3} - 14{x^2} - x + 6$
When $x = \dfrac{3}{2}$,
$
   \Rightarrow 8{\left( {\dfrac{3}{2}} \right)^3} - 14{\left( {\dfrac{3}{2}} \right)^2} - \dfrac{3}{2} + 6 \\
   \Rightarrow 8\left( {\dfrac{{27}}{8}} \right) - 14\left( {\dfrac{9}{4}} \right) - \dfrac{3}{2} + 6 \\
   \Rightarrow 27 - \dfrac{{63}}{2} - \dfrac{3}{2} + 6 \\
   \Rightarrow \dfrac{{54 - 63 - 3 + 12}}{2} \\
   \Rightarrow \dfrac{0}{2} \\
   \Rightarrow 0 \\
 $
When $x = - \dfrac{7}{5}$
$
   \Rightarrow 8{\left( { - \dfrac{7}{5}} \right)^3} - 14{\left( { - \dfrac{7}{5}} \right)^2} - \left( { - \dfrac{7}{5}} \right) + 6 \\
   \Rightarrow 8\left( { - \dfrac{{343}}{{125}}} \right) - 14\left( {\dfrac{{49}}{{25}}} \right) + \dfrac{7}{5} + 6 \\
   \Rightarrow - \dfrac{{2744}}{{125}} - \dfrac{{686}}{{25}} + \dfrac{7}{5} + 6 \\
   \Rightarrow \dfrac{{ - 2744 - 3430 + 175 + 750}}{{125}} \\
   \Rightarrow \dfrac{{ - 5249}}{{125}} \\
 $
Therefore, it is clearly visible that $x = \dfrac{3}{2}$ divides $8{x^3} - 14{x^2} - x + 6$ completely.
So, HCF =$2x - 3$

Hence, the correct answer is option (A).

Note: The concept of HCF is easy to understand. Some of the important terms which students should know while solving these types of questions are- factors, prime factors, factor tree and common factor. The easiest way to solve HCF related questions is by creating a factor tree. Students can practise more similar types of questions for better understanding and clarity. The solutions of HCF questions can easily be verified using the other methods. Let us say, you solve a question using the prime factorization method. The solution can be verified by a long division method.