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# ., p[1] = c1, and p[0] = c0.

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In this problem, you are asked to implement polynomials

using a list. A polynomial is an expression of the form

c0 + c1x + c2x2 + c3x3 + . . . + cnxn

where c0, c1, . . . , cn are real numbers. Each of cixi is called a term of the polynomial, and ci

is called the coefficient of the term with exponent i. Note that c0 is simply the coefficient of

the term with exponent 0. The maximum exponent with a non-zero coefficient is called the

degree of the polynomial. For example, 6×14 + 9×11 −12×3 + 42 is a polynomial of degree 14.

The polynomial −12×6 + 5×5 −20×4 + 8×2 −12x + 9 has degree 6. The following are some

standard operations on polynomials:

3

Scaling a polynomial: Given a polynomial p(x) = cnxn+cn−1xn−1+. . .+c1x+c0 and a real

value s, scaling p(x) by s gives the polynomial s ·p(x) obtained by scaling the coefficient

of every term in p(x) by the factor s. For example, if p(x) = 6×14 + 9×11 −12×3 + 42,

then 2p(x) = 12×14 + 18×11 −24×3 + 84.

Sum of two polynomials: The sum of two polynomials p1(x) and p2(x), denoted by p1(x)+

p2(x), is the polynomial obtained by adding the terms of p1(x) and p2(x). For example,

if p1(x) = 6×14 + 9×11 −12×3 + 42 and p2(x) = −12×6 + 5×5 −20×4 + 8×2 −12x + 9, then

p1(x) + p2(x) = 6×14 + 9×11 −12×6 + 5×5 −20×4 −12×3 + 8×2 −12x + 51.

Difference of two polynomials: The difference of two polynomials p1(x) and p2(x) is the

polynomial obtained by subtracting one from the other. Therefore, p1(x) −p2(x) is the

polynomial obtained by subtracting the terms of p2(x) from p1(x).

For example, if p1 and

p2 are as above, then p1(x)−p2(x) = 6×14+9×11+12×6−5×5+20×4−12×3−8×2+12x+33.

Product of two polynomials: The product of two polynomials p1(x) and p2(x), denoted

by p1(x) ·p2(x), is the polynomial obtained by the pair-wise multiplication of terms in

p1(x) and p2(x). For example, if p1(x) = x4 + 4×3 + 4×2 and p2(x) = 2x −1, then

p1(x) ·p2(x) = 2×5 + 7×4 + 4×3 −4×2. This is obtained as follows: (x4 + 4×3 + 4×2)(2x) +

(x4 + 4×3 + 4×2)(−1) = 2×5 + 8×4 + 8×3 −x4 −4×3 −4×2 = 2×5 + 7×4 + 4×3 −4×2.

In this program, you will store a polynomial as a list of real values. The length of the list will

be equal to one more than the degree of the polynomial. The item at index i of the list will

be equal to the coefficient of the term with exponent i in the polynomial. In other words,

polynomial p(x) = cnxn + cn−1xn−1 + . . . + c1x + c0 is stored in a list p of length n + 1, such

that p[n] = cn, p[n-1] = cn−1, . . ., p[1] = c1, and p[0] = c0. For example, the polynomial

6×14 + 9×11 −12×3 + 42 is stored as the following list: [42, 0, 0, -12, 0, 0, 0, 0, 0,

0, 0, 9, 0, 0, 6]. Note that the length of the list is 15.

You are asked to implement the following functions to manipulate polynomials:

1. A function called make poly with a single parameter called termlist, which

is a list of 2-tuples, where each 2-tuple in the list is an (exponent, coefficient) pair

representing a term in the polynomial. The exponent is an integer and the coefficient

is a real number. This function will create a list to store the polynomial (as described

above) and return the list. Note that the 2-tuples in termlist may appear in any order

(that is, do not assume that the terms will appear in increasing or decreasing order of

exponents in termlist). For example, after the following assignment statement

term_list_1 = [(14, 6), (11, 9), (0, 42), (3, -12)]

p1 = make_poly(term_list_1)

p1 should be the list [42, 0, 0, -12, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 6].

2. function called print poly with a single parameter: a polynomial p. Keep

in mind that p is a list. This function prints out the polynomial. The polynomial must be

printed “nicely”. This means that only terms with non-zero coefficients must be printed

out and the terms must be printed from largest exponent to the smallest. Furthermore, if

the coefficient of a term is negative, the – (minus sign) must be embedded in the printed

polynomial. For example, if the polynomial is −12×6 + 5×5 −20×4 + 8×2 −12x + 9, it

should be printed as

-12x^6 + 5x^5 – 20x^4 + 8x^2 – 12x + 9

4

and not as

-12x^6 + 5x^5 + -20x^4 + 0x^3 + 8x^2 + -12x^1 + 9x^0

3. A function called eval poly with two parameters: a polynomial p and a real

number r. The function should evaluate p at x = r and return the result. For example,

if p is the polynomial −12×6 + 5×5 −20×4 + 8×2 −12x + 9, then eval poly(p, 2) should

return -911 (which we obtain by plugging in 2 as the value of x).

4. A function called scale poly with two parameters: a polynomial p and a real

non-zero value s. The function should return the polynomial sp, that is, the polynomial

obtained by scaling p by s. Keep in mind that you are returning a new polynomial (that

is, this function should not modify p).

5. A function called sum poly with two parameters: a polynomial p1 and a

polynomial p2. The function should return the sum polynomial p1 + p2. Make sure

that the length of the sum polynomial list is exactly equal to its degree plus one. Also,

the function should not modify p1 and p2.

6. A function called diff poly with two parameters: a polynomial p1 and a

polynomial p2. The function should return the difference polynomial p1 – p2. Again,

make sure that the length of the difference polynomial list is exactly equal to its degree

plus one and the function does not modify p1 and p2.

7. A function called mult poly with two parameters: a polynomial p1 and a

polynomial p2. The function should return the product polynomial p1 * p2. Make sure

that the length of the product polynomial list is exactly equal to its degree plus one and

the function does not modify p1 and p2.

8. A function called test polys without any parameters. This function has been writ-

ten for you in a file called hw4problem2.py, available for download on Canvas.

This function is simply a driver function that makes calls to all of the above functions,

thus allowing you to test them. You will insert your implementations of all the above

functions into this module.

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