Question

How do you use synthetic division to show that $x = 2$ is a zero of ${x^3} - 7x + 6 = 0$ ?

Answer 1

In order to this question, to show that $x = 2$ is a zero of ${x^3} - 7x + 6 = 0$ by using synthetic division, we will go through few steps to use the whole synthetic division method, as we will start division by the divisor 2 because $x = 2$ is given, and then we will divide step by step each coefficient until the last coefficient is divided. And if the remainder remains 0, then $x = 2$ is a zero of ${x^3} - 7x + 6 = 0$ .

Complete answer: To divide the given equation ${x^3} - 7x + 6 = 0$ or ${x^3} + 0{x^2} - 7x + 6$ by $x = 2$ :
As we have to show $x = 2$ as a zero of the given equation, it is possible if the given equation is divisible by $x = 2$ .
Now, we will follow few steps to complete the synthetic division:-
Step-1: Write the coefficients of $x$ in the dividend inside an upside-down division symbol.
|\,\,1\,\,\,\,0\,\,\,\, - 7\,\,\,\,6 \\\ |\,\\_\,\\_\,\\_\,\\_\,\\_\,\\_\,\\_\,\\_\, \\\
Step-2: As $x = 2$ , we put 2 at the left.
2|\,\,1\,\,\,\,0\,\,\,\, - 7\,\,\,\,6 \\\ \,\,\,\,|\,\\_\,\\_\,\\_\,\\_\,\\_\,\\_\,\\_\,\\_ \\\
Step-3: Remove the first coefficient of the dividend below the division symbol.
2|\,\,1\,\,\,\,0\,\,\,\, - 7\,\,\,\,6 \\\ \,\,\,\,|\,\\_\,\\_\,\\_\,\\_\,\\_\,\\_\,\\_\,\\_\, \\\ \,\,\,\,|\,\,1 \\\
Step-4: Multiply the result by the constant, and put the product in the next column.
2|\,\,1\,\,\,\,0\,\,\,\, - 7\,\,\,\,6 \\\ \,\,\,\,|\,\\_\,\\_\,2\\_\,\\_\,\\_\,\\_\,\\_\,\\_\, \\\ \,\,\,\,|\,\,1 \\\
Step-5: Add down the column.
2|\,\,1\,\,\,\,0\,\,\,\, - 7\,\,\,\,6 \\\ \,\,\,\,|\,\\_\,\\_\,2\\_\,\\_\,\\_\,\\_\,\\_\,\\_\, \\\ \,\,\,\,|\,\,1\,\,\,\,2 \\\
Step-6: Repeat step 4 and 5 until we can go no farther.
2|\,\,1\,\,\,\,0\,\,\,\, - 7\,\,\,\,6 \\\ \,\,\,\,|\,\\_\,\\_\,2\\_\,\,4\,\\_\, - 6\,\\_\,\\_\, \\\ \,\,\,\,|\,\,1\,\,\,\,2\,\,\,\,\,\, - 3\,\,\,\,0 \\\
Hence, Quotient is ${x^2} + 2x - 1$ and the remainder is 0.
And as the remainder is 0, $x = 2$ is a zero of ${x^3} - 7x + 6 = 0$ .

Note:
Synthetic division is a shorthand, or shortcut, form of polynomial division that only works in the exceptional scenario of dividing by a linear factor. Synthetic division is most commonly employed for obtaining polynomial zeros (or roots) rather than dividing out factors.