Polynomial evaluation and roots in MATLAB - Step-by-Step Execution

Concept Flow - Polynomial evaluation and roots

Define polynomial coefficients

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Evaluate polynomial at x

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Calculate polynomial value

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Find roots of polynomial

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Output roots and values

Start by defining polynomial coefficients, then evaluate the polynomial at a point x, calculate its value, find roots, and output results.

Execution Sample

MATLAB

p = [1 -3 2];
x = 2;
y = polyval(p, x);
r = roots(p);

Defines polynomial p(x) = x^2 - 3x + 2, evaluates it at x=2, and finds its roots.

Execution Table

Step	Action	Expression	Result
1	Define polynomial coefficients	p = [1 -3 2]	[1 -3 2]
2	Set evaluation point	x = 2	2
3	Evaluate polynomial at x	y = polyval(p, x)	0
4	Calculate roots of polynomial	r = roots(p)	[1; 2]
5	Output results	y and r	y=0, r=[1; 2]

💡 All steps completed; polynomial evaluated and roots found.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	Final
p	undefined	[1 -3 2]	[1 -3 2]	[1 -3 2]	[1 -3 2]	[1 -3 2]
x	undefined	undefined	2	2	2	2
y	undefined	undefined	undefined	0	0	0
r	undefined	undefined	undefined	undefined	[1; 2]	[1; 2]

Key Moments - 2 Insights

Why does polyval(p, 2) return 0 even though the polynomial looks complex?

Because at x=2, the polynomial x^2 - 3x + 2 equals zero, as shown in execution_table step 3 where y=0.

Why are the roots returned as [2; 1] and not in ascending order?

MATLAB roots function returns roots in no guaranteed order; here roots are 1 and 2 as in execution_table step 4.

Visual Quiz - 3 Questions

Look at the execution_table, what is the value of y after step 3?

A 2

B 1

C 0

D -1

💡 Hint

Check the 'Result' column in row with Step 3 in execution_table.

At which step are the roots of the polynomial calculated?

A Step 4

B Step 2

C Step 3

D Step 5

💡 Hint

Look at the 'Action' column in execution_table for root calculation.

If x was changed to 1, what would be the new value of y in the variable_tracker?

A 1

B 0

C 2

D Undefined

💡 Hint

Evaluate polynomial at x=1: 1^2 - 3*1 + 2 = 0; relate to variable_tracker for y.

Concept Snapshot

Polynomial evaluation and roots in MATLAB:
- Define polynomial as vector of coefficients, highest power first.
- Use polyval(p, x) to evaluate polynomial at x.
- Use roots(p) to find polynomial roots.
- Roots may be real or complex.
- Example: p=[1 -3 2] means x^2 -3x +2.

Full Transcript

This example shows how to define a polynomial in MATLAB using a vector of coefficients. We then evaluate the polynomial at a specific value x using the polyval function. Next, we find the roots of the polynomial using the roots function. The execution table traces each step, showing variable values and results. Key moments clarify why the polynomial evaluates to zero at x=2 and why roots order is not guaranteed. The visual quiz tests understanding of these steps. The concept snapshot summarizes the key commands and behavior for polynomial evaluation and root finding in MATLAB.