exam1.html

MCS 320 Exam 1 Wednesday 21 February 2007

1. Consider the polynomial Give Maple commands (with output)

> p := x^3 + 3*x^2 + 8;

(a) to show that p is irreducible over the rational numbers :

(b) to factor p symbolically into a product of linear polynomials, adding sufficiently many formal roots :

(c) to factor p numerically :

(d) to show the relation between the numerical and the symbolic factorization :

2. Execute the commands

> restart; k := 3; sum(x[k],k=1..5);

Error, (in sum) summation variable previously assigned, second argument evaluates to 3 = 1 .. 5

(a) Explain why the third command does not give the desired result.

The variable k has already a value before the sum command is executed.

The sum command can therefore not use k to evaluate the range 1..5.

(b) Modify ONLY the third command so that is returns

> sum(x['k'],'k'=1..5);

3. The sequence

gives rational approximations to

When answering the questions below, also give the Maple commands used:

(a) How many decimal places in as rational approximation for are correct?

> s2 := evalf(sqrt(2));

> r := evalf(66441/46981);

We see that six decimal places agree.

(b) What is the next element in the sequence?

As the previous number is correct up to six decimal places, we evaluate sqrt(2) with 7 decimal places before converting it to a rational number:

> convert(evalf(sqrt(2),7),rational);

4. Do q := (x/y - 1)/(y/x - 1); and consider the expression assigned to q.

> q := (x/y - 1)/(y/x - 1);

(a) Give the Maple command(s) to display the internal representation of this expression.

PROD(135277496,5)
  SUM(135277424,5)
     PROD(135277400,5)
        NAME(135011340,4): x
        INTPOS(3,2): 1
        NAME(135232876,4): y
        INTNEG(-1,2): -1
     INTPOS(3,2): 1
     INTNEG(-1,2): -1
     INTPOS(3,2): 1
  INTPOS(3,2): 1
  SUM(135277472,5)
     PROD(135277448,5)
        NAME(135232876,4): y
        INTPOS(3,2): 1
        NAME(135011340,4): x
        INTNEG(-1,2): -1
     INTPOS(3,2): 1
     INTNEG(-1,2): -1
     INTPOS(3,2): 1
  INTNEG(-1,2): -1

(b) Draw the directed acyclic graph which represents this expression.

See the handout on paper.

(c) Explain the outcome of subs(-1=+2,q). Why are all the divisions gone?

Maple represents every object only once. Not only do the coefficients "-1" turn into "+2", but also the divisions disappear, because a division is represented as a power product with a negative exponent for the denominator.

5. Let . Give all Maple commands and their output when answering the questions below.

> r := (x^14-8*x^12+9*x)/(x^13+14*x^7-3*x^5); 1

(a) Convert into a Horner form.

(b) Compare to the given form of , how many arithmetical operations are gained when using the Horner form of to evaluate

We gained 28 multiplications at the expense of one division.

(c) What is the most efficient way to evaluate ?

     t1 = x*x;
     t2 = t1*t1;
     t4 = t2*t2;
     t10 = t2*x;
     t18 = (t4*t2*t1-8.0*t4*t2+9.0*x)/(t4*t10+14.0*t2*t1*x-3.0*t10);

Here we count only 14 multiplications, 4 additions, and one division.

6. Consider the polynomials and

(a) Are these polynomials the same or are they different?

> p := x^2 - 7/4; q := x^2 - 1.75;

They look different, but

they are only the same if 1.75 = 7/4. In computer algebra 7/4 is an exact rational number and consequently p is a polynomial with coefficients in the rational number field, whereas 1.75 is regarded as a floating-point number of limited precision, here by default 10. So the polynomial q is seen by Maple as a polynomial with floating-point real coefficients.

(b) Explain why the outcome of factor is then so different.

The polynomial p has rational coefficients, we call it a rational polynomial.

The polynomial p has real floating-point coefficients, we call it a real polynomial.

The rational polynomial p has no rational roots, so it does not factor over the rational numbers.

The real polynomial q has two real roots, so it factors.