Answer :
To apply synthetic division to a polynomial, set up an array with the coefficients and constant of the polynomial and the test value c. Carry out the synthetic division operation, and if the final result is 0, c is a root of the polynomial. If not, the value of the polynomial at x = c is the final result.
The question asks for the use of synthetic division to determine if a given value c is a zero of a polynomial function. To illustrate with one of the examples provided, for the polynomial function p(x) = 32x³ - 80x² + 80x - 40 with c = 1 :
Arrange the coefficients and constant term of the polynomial in an array, (32 -80 80 -40), followed by the value c (which is 1 for this case). Draw a line under the array and bring down the first number (32) below the line. Multiply c by the number you brought down, place the result beneath the next number in the array, and add the numbers in that column. Repeat this step until you reach the end of the row. The final number you obtain is the remainder. If it is 0, c is a root of the polynomial. If it is not 0, it is the value of the polynomial at x = c, i.e., p(c).
In this case, the remainder is not zero, indicating that 1 is not a root of the polynomial. The value p(1) is the remainder, which is -8.
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