The product of 3 consecutive integers is -1716. What is the greatest of the 3 integers.

Question

Please show it in an equation and explain a little more when and how synthetic division is used in that equation.

Answer 1

Let x, x + 1, x + 2 be the three integers

 

Then, x(x + 1)(x + 2) = -1716

 

x(x² + 3x + 2) = -1716

 

x³ + 3x² + 2x + 1716 = 0

 

By trial and error, -13 is a root:

 

-13⌋ 1      3     2       1716

                     130    -1716             

       1    -10    132      0

 

x³ + 3x² + 2x + 1716 = (x + 13)(x² -10x + 132)

 

The roots of x² - 10x + 132 are not real numbers

 

So, x = -13

 

The numbers are -13, -12, -11

 

Answer 2

1st integer = x

2nd integer = x+1

3rd integer = x+2

 

Your equation is

 

x(x+1)(x+2) = -1716

 

x(x²+3x+2) = -1716

 

x³+2x²+2x = -1716

 

x³+2x²+2x+1716 = 0

 

 

Synthetic division is used in order to find a factor binomial that can go into the polynomial.  For instance, if you divide the polynomial by the factor binomial, you should obtain a quotient with remainder zero.  Synthetic division uses the coefficients of the terms by multiplying the coefficients of the binomial & polynomial and adding them.  If the sum of the last coefficient on the right side is zero, you have found your other factor.  You can then use those factors to find the roots of the polynomial.