Polynomial operations with np.poly in NumPy - Step-by-Step Execution

Concept Flow - Polynomial operations with np.poly

Define polynomial coefficients

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Create polynomial object with np.poly1d

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Perform operations: add, multiply, evaluate

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Get results: new polynomial coefficients or values

Start by defining polynomial coefficients, create polynomial objects, then perform operations like addition, multiplication, or evaluation to get new polynomials or values.

Execution Sample

NumPy

import numpy as np
p1 = np.poly1d([1, 2, 3])
p2 = np.poly1d([3, 4])
sum_poly = p1 + p2
prod_poly = p1 * p2
val = p1(2)

This code creates two polynomials, adds and multiplies them, and evaluates one at x=2.

Execution Table

Step	Action	Polynomial/Value	Result
1	Create p1 with coefficients [1, 2, 3]	p1	p1 = 1*x^2 + 2*x + 3
2	Create p2 with coefficients [3, 4]	p2	p2 = 3*x + 4
3	Add p1 + p2	sum_poly	sum_poly = 1*x^2 + 5*x + 7
4	Multiply p1 * p2	prod_poly	prod_poly = 3*x^3 + 10*x^2 + 17*x + 12
5	Evaluate p1 at x=2	val	val = 11
6	End of operations		All operations complete

💡 All polynomial operations done and evaluated.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	After Step 5	Final
p1	undefined	poly1d([1, 2, 3])	poly1d([1, 2, 3])	poly1d([1, 2, 3])	poly1d([1, 2, 3])	poly1d([1, 2, 3])	poly1d([1, 2, 3])
p2	undefined	undefined	poly1d([3, 4])	poly1d([3, 4])	poly1d([3, 4])	poly1d([3, 4])	poly1d([3, 4])
sum_poly	undefined	undefined	undefined	poly1d([1, 5, 7])	poly1d([1, 5, 7])	poly1d([1, 5, 7])	poly1d([1, 5, 7])
prod_poly	undefined	undefined	undefined	undefined	poly1d([3, 10, 17, 12])	poly1d([3, 10, 17, 12])	poly1d([3, 10, 17, 12])
val	undefined	undefined	undefined	undefined	undefined	11	11

Key Moments - 3 Insights

Why does adding p1 and p2 result in a polynomial of degree 2, not 3?

Because p1 is degree 2 and p2 is degree 1, adding them aligns terms by degree. The highest degree term comes from p1, so sum_poly remains degree 2 as shown in step 3 of the execution_table.

How does multiplication increase the polynomial degree?

Multiplying p1 (degree 2) by p2 (degree 1) adds their degrees, resulting in degree 3. This is shown in step 4 where prod_poly has degree 3.

What does p1(2) mean and how is it calculated?

p1(2) means evaluate polynomial p1 at x=2. It calculates 1*2^2 + 2*2 + 3 = 4 + 4 + 3 = 11, as shown in step 5.

Visual Quiz - 3 Questions

Look at the execution_table, what is the coefficient of x in sum_poly after step 3?

A 2

B 4

C 5

D 3

💡 Hint

Check the 'Polynomial/Value' column in step 3 of execution_table.

At which step does the polynomial degree increase to 3?

A Step 4

B Step 3

C Step 2

D Step 5

💡 Hint

Look at the 'Result' column for degree info in execution_table rows.

If p1 was evaluated at x=3 instead of x=2, what would val be?

A 12

B 18

C 15

D 20

💡 Hint

Use the formula from key_moments step 3 with x=3.

Concept Snapshot

np.poly1d creates polynomial objects from coefficient lists.
You can add (+) and multiply (*) polynomials directly.
Evaluate polynomials by calling them like functions, e.g., p(2).
Addition aligns terms by degree; multiplication adds degrees.
Coefficients are ordered from highest degree to constant term.

Full Transcript

This visual execution shows how to use numpy's np.poly1d to work with polynomials. First, we define polynomials by their coefficients. Then, we create polynomial objects. We can add or multiply these polynomials, which changes their coefficients and degree accordingly. Finally, we can evaluate a polynomial at a specific value of x by calling it like a function. The execution table traces each step, showing how variables change and what results are produced. Key moments clarify common confusions about polynomial degree and evaluation. The quiz tests understanding by referencing the execution steps and variable states.