Polynomial operations with np.poly in NumPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of polynomial addition using np.polyadd

What is the output of this code that adds two polynomials using np.polyadd?

NumPy

import numpy as np
p1 = np.array([1, -3, 2])  # x^2 - 3x + 2
p2 = np.array([0, 4, -1])  # 4x - 1
result = np.polyadd(p1, p2)
print(result)

A [ 1 -3 6 1]

B [ 1 1 1]

C ]1 1 1 [

D 1 1 1]

💡 Hint

Remember that np.polyadd adds coefficients of the same degree terms.

The polynomials are p1 = x^2 - 3x + 2 and p2 = 4x - 1. Adding them gives x^2 + ( -3 + 4 )x + (2 - 1) = x^2 + x + 1, so coefficients are [1, 1, 1].

❓ data_output

intermediate

Degree of polynomial from roots using np.poly

Given roots [-1, 0, 2], what is the degree and coefficients of the polynomial generated by np.poly?

NumPy

import numpy as np
roots = [-1, 0, 2]
coeffs = np.poly(roots)
degree = len(coeffs) - 1
print(degree)
print(coeffs)

A

3
[ 1. -1.  2.  0.]

B

].0  .2- .1- .1 [
3

C

3
[ 1. -1. -2.  0.]

D

].0  .2  .1- .1 [
3

💡 Hint

np.poly returns coefficients of polynomial with given roots, highest degree first.

The polynomial with roots -1, 0, 2 is (x+1)(x)(x-2) = x^3 - x^2 - 2x. Coefficients are [1, -1, -2, 0]. Degree is 3.

❓ visualization

advanced

Plot polynomial and its derivative

Which option correctly plots the polynomial p(x) = 2x^3 - 3x^2 + x - 5 and its derivative using np.polyder?

NumPy

import numpy as np
import matplotlib.pyplot as plt
p = np.array([2, -3, 1, -5])
p_der = np.polyder(p)
x = np.linspace(-2, 3, 100)
y = np.polyval(p, x)
y_der = np.polyval(p_der, x)
plt.plot(x, y, label='Polynomial')
plt.plot(x, y_der, label='Derivative')
plt.legend()
plt.show()

A Plots polynomial and derivative curves correctly with labels.

B Plots only polynomial curve, derivative missing.

C Plots derivative curve only, polynomial missing.

D Raises error due to wrong np.polyder usage.

💡 Hint

np.polyder returns coefficients of the derivative polynomial.

Option A correctly uses np.polyder and plots both polynomial and its derivative with labels.

🔧 Debug

advanced

Identify error in polynomial multiplication code

What error does this code raise when multiplying two polynomials using np.polymul incorrectly?

NumPy

import numpy as np
p1 = [1, 2]
p2 = (3, 4)
result = np.polymul(p1, p2)
print(result)

A TypeError: polymul() missing 1 required positional argument

B SyntaxError: invalid syntax

C No error, outputs [3, 10, 8]

D TypeError: polymul() argument 2 must be array-like, not tuple

💡 Hint

Check the data types of the polynomial coefficient inputs.

p2 is a tuple, but np.polymul expects array-like (list or np.array). Passing a tuple causes TypeError.

🚀 Application

expert

Find roots of polynomial from coefficients

Given polynomial coefficients [1, -6, 11, -6], which option correctly finds all roots using np.roots?

NumPy

import numpy as np
coeffs = [1, -6, 11, -6]
roots = np.roots(coeffs)
print(sorted(roots))

A [1.0, 2.0, 3.0]

B [-3.0, 1.0, 2.0]

C [3.0, 2.0, 1.0]

D Raises ValueError due to invalid coefficients

💡 Hint

np.roots returns roots in any order; sorting helps compare.

Polynomial is (x-1)(x-2)(x-3), roots are 1, 2, 3. Sorted roots are [1.0, 2.0, 3.0].

Practice

1. What does the np.poly function do in NumPy?

easy

A. It evaluates a polynomial at given points.

B. It calculates the derivative of a polynomial.

C. It finds the roots of a polynomial from coefficients.

D. It returns polynomial coefficients from given roots.

Solution

1. Step 1: Understand the purpose of np.poly

   The function np.poly takes roots of a polynomial as input and returns the coefficients of the polynomial with those roots.

2. Step 2: Differentiate from other polynomial functions

   Other functions like np.polyder find derivatives, np.polyval evaluates polynomials, and np.roots finds roots from coefficients, so they are different.

3. Final Answer:

   It returns polynomial coefficients from given roots. -> Option D

4. Quick Check:

   np.poly = coefficients from roots [OK]

Hint: Remember: np.poly converts roots to coefficients [OK]

Common Mistakes:

• Confusing np.poly with np.roots
• Thinking np.poly evaluates polynomials
• Mixing up derivative functions with np.poly

2. Which of the following is the correct syntax to get polynomial coefficients from roots [-1, 2, 3] using NumPy?

easy

A. np.poly([-1, 2, 3])

B. np.polyroots([-1, 2, 3])

C. np.polyval([-1, 2, 3])

D. np.polycoef([-1, 2, 3])

Solution

1. Step 1: Recall the correct function name

   The function to get polynomial coefficients from roots is np.poly.

2. Step 2: Check the syntax

   The correct syntax is np.poly([-1, 2, 3]). Other options are not valid NumPy functions.

3. Final Answer:

   np.poly([-1, 2, 3]) -> Option A

4. Quick Check:

   Correct function is np.poly with roots list [OK]

Hint: Use np.poly with a list of roots inside parentheses [OK]

Common Mistakes:

• Using non-existent functions like np.polyroots
• Confusing np.polyval with np.poly
• Misspelling function names

3. What is the output of the following code?

import numpy as np
coeffs = np.poly([1, 2])
print(coeffs)

medium

A. [1. 3. 2.]

B. [1. -3. 2.]

C. [1. -2. 1.]

D. [1. 2. -3.]

Solution

1. Step 1: Understand np.poly output for roots [1, 2]

   np.poly returns coefficients of polynomial with roots 1 and 2, which is (x - 1)(x - 2) = x^2 - 3x + 2.

2. Step 2: Write coefficients in descending order

   The coefficients are [1, -3, 2], representing 1*x^2 - 3*x + 2.

3. Final Answer:

   [1. -3. 2.] -> Option B

4. Quick Check:

   Roots [1,2] -> coeffs [1, -3, 2] [OK]

Hint: Multiply (x - root) terms and list coefficients [OK]

Common Mistakes:

• Mixing signs of coefficients
• Listing roots instead of coefficients
• Confusing order of coefficients

4. The following code throws an error. What is the problem?

import numpy as np
coeffs = np.poly(3, 4, 5)
print(coeffs)

medium

A. roots should be a list or array, not a tuple without brackets

B. np.poly does not accept more than two roots

C. print statement syntax is incorrect

D. np.poly requires integer roots only

Solution

1. Step 1: Check the input to np.poly

   The roots 3, 4, 5 are passed as separate positional arguments instead of a single sequence. np.poly expects one array-like input (list, array, or tuple).

2. Step 2: Fix the roots input

   Wrap roots in brackets: coeffs = np.poly([3, 4, 5]) to pass a list to np.poly.

3. Final Answer:

   roots should be a list or array, not a tuple without brackets -> Option A

4. Quick Check:

   np.poly needs list/array input [OK]

Hint: Always pass roots as list or array to np.poly [OK]

Common Mistakes:

• Passing tuple without brackets
• Assuming np.poly limits root count
• Misunderstanding print syntax errors

5. You have roots [-2, 0, 3] and want to create a polynomial with leading coefficient 2 instead of 1. How can you do this using np.poly?

hard

A. Add 2 to the coefficients returned by np.poly

B. Pass an extra argument 2 to np.poly as np.poly([-2, 0, 3], 2)

C. Multiply the output of np.poly([-2, 0, 3]) by 2

D. Use np.polyval with 2 as the first argument

Solution

1. Step 1: Generate coefficients from roots

   Use np.poly([-2, 0, 3]) to get coefficients for polynomial with leading coefficient 1.

2. Step 2: Scale coefficients to change leading coefficient

   Multiply all coefficients by 2 to make leading coefficient 2, since np.poly does not take leading coefficient as argument.

3. Final Answer:

   Multiply the output of np.poly([-2, 0, 3]) by 2 -> Option C

4. Quick Check:

   Scale coefficients to change leading term [OK]

Hint: Multiply coefficients array to adjust leading coefficient [OK]

Common Mistakes:

• Trying to pass leading coefficient as np.poly argument
• Adding instead of multiplying coefficients
• Using np.polyval incorrectly