What is the result of dividing the polynomial px = x^4 + 2x^3 + 3x^2 + ax + 3a + 7 by x – 1?

Dividing the Polynomial

To divide the polynomial p(x) = x⁴ + 2x³ + 3x² + ax + 3a + 7 by x – 1, we can apply polynomial long division or synthetic division. Here, we will use synthetic division for simplicity.

Step-by-Step Solution

1. Set up synthetic division:

   We will substitute x = 1 (as x – 1 = 0 when x = 1) into the polynomial and set up the corresponding coefficients:

   • Coefficients: 1 (for x⁴), 2 (for x³), 3 (for x²), a (for x), 3a (constant term), and 7.
2. Perform synthetic division:

               1 | 1   2   3   a   3a   7
                 |     1   3   6  a+6  3a+a+1
               
                 --------------------------
                  1   3   6   a + 3a + 7
               

   The numbers directly below the line represent the coefficients of the divisor.

3. Result:

   The result from the synthetic division is:

               p(x) = (x3 + 3x2 + 6x + (4a + 7)) + R,
               

   where R = 4a + 7 is the remainder.

Conclusion

Therefore, when dividing the polynomial p(x) by x – 1, the resulting polynomial is x³ + 3x² + 6x + (4a + 7), with a remainder of 4a + 7.