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Exercises

Loops and branching

Write a program that can determine if a (positive) number is prime. Fill out this skeleton program:

/* primetest.c */

#include <stdio.h>

int main(int argc, char *argv[])
{
  unsigned int number = 51;

  /*
   * Insert code here
   */

  return 0;
}

Your program should print out

$ make
gcc -std=c99 -Wall primetest.c -o primetest.exe
$ ./primetest.exe
51 is not prime

You can test if a number \(N\) is prime by checking if it is divisible by any number \(n\) where \(2 \leq n\) and \(n^2 \leq N\). Try changing the value of number to check that it works. Now write a program that can print out all the prime numbers less than 100.

Write a program that can check if an integer \(x\) is a root of a cubic polynomial \[ax^3 + bx^2 + cx + d\] where \(a\), \(b\), \(c\) and \(d\) are integers. Fill out the skeleton

/* isroot.c */

#include <stdio.h>

int main(int argc, char *argv[])
{
  int x = 5;

  int a = 1;
  int b = -4;
  int c = -7;
  int d = 10;

  /*
   * Insert code here
   */

  return 0;
}

Your program should print out something like

$ make
gcc -std=c99 -Wall isroot.c -o isroot.exe
$ ./isroot.exe
5 is a root of 1x^3 + -4x^2 + -7x + 10

Try different values for x, a, etc. Feel free to try and make the output polynomial look better…

Cubic solver

The rational root theorem states that any integer root of a cubic with integer coefficients divides \(d\). Hence any integer root \(x\) must satisfy \(-d \le x \le d\) (assuming that \(d\) is positive). Adapt the above program so that it prints out all integer roots of the given cubic. Test your program out with different cubics. Example output might be

$ ./findroots.exe
Roots of 1x^3 + -4x^2 + -7x + 10:
-2 1 5

Functions

Take the programs you wrote for the previous loop exercises and split them up using functions. In particular create a function isprime(n) that returns whether or not an integer n is prime.

Also make a function polyeval(x, a, b, c, d) that can evaluate the polynomial \[P(x) = ax^3 + bx^2 + cx + d.\] Then use that function to make a function isroot(x, a, b, c, d) that returns whether or not x is a root of \(P(x)\). Finally make a function polyprint that can print a polynomial to the terminal. Use all of these functions in your previous program to find the integer roots of a cubic. Think carefully about the output of the polyprint function. Does it print all polynomials the way you want it to?