Polynomial evaluation and roots in MATLAB - Cheat Sheet & Quick Revision

Recall & Review

beginner

What MATLAB function is used to evaluate a polynomial at specific points?

The polyval function evaluates a polynomial at given points. You provide the polynomial coefficients and the points where you want to evaluate it.

beginner

How do you find the roots of a polynomial in MATLAB?

Use the roots function with the polynomial coefficients as input. It returns the values where the polynomial equals zero.

beginner

Given polynomial coefficients p = [1 -3 2], what polynomial does this represent?

It represents the polynomial x² - 3x + 2. The vector lists coefficients from highest degree to constant term.

intermediate

What is the output of roots([1 -3 2]) in MATLAB?

The output is [2; 1], which are the roots of the polynomial x² - 3x + 2.

beginner

How can you evaluate the polynomial x² - 3x + 2 at x = 5 in MATLAB?

Use polyval([1 -3 2], 5). This calculates 5² - 3*5 + 2 = 25 - 15 + 2 = 12.

Which MATLAB function returns the roots of a polynomial given its coefficients?

A polyfit

B polyval

C rootsolve

D roots

The roots function finds the zeros of the polynomial.

What does the vector [2 -4 1] represent in MATLAB polynomial terms?

A 2x² - 4x + 1

B 2 + -4x + x²

C x² - 4x + 2

D 1x² - 4x + 2

Coefficients are listed from highest degree to constant term.

What will polyval([1 0 -1], 1) return?

A 0

B 1

C -1

D 2

Evaluates x² - 1 at x=1: 1² - 1 = 0.

If a polynomial has coefficients [1 -2 1], what are its roots?

A [-1; -1]

B [1; 1]

C [0; 1]

D [2; -1]

The polynomial x² - 2x + 1 factors as (x-1)², root at 1.

Which MATLAB function would you use to fit a polynomial to data points?

A roots

B polyval

C polyfit

D polyroot

polyfit fits a polynomial to data, not for evaluation or roots.

Explain how to represent a polynomial in MATLAB and how to evaluate it at a given point.

Think about how you write the polynomial as a list of numbers and then use a function to find its value.

Describe the process to find the roots of a polynomial using MATLAB and what the roots represent.

Roots are where the polynomial equals zero.

Practice

1. What does the MATLAB function polyval(p, x) do when p is a vector of polynomial coefficients and x is a number?

easy

A. Evaluates the polynomial at the value x.

B. Finds the roots of the polynomial p.

C. Plots the polynomial curve over x.

D. Calculates the derivative of the polynomial at x.

Solution

1. Step 1: Understand the function inputs

   The vector p contains polynomial coefficients, and x is the point where we want to evaluate the polynomial.

2. Step 2: Identify the function purpose

   polyval calculates the polynomial value at x using the coefficients in p.

3. Final Answer:

   Evaluates the polynomial at the value x. -> Option A

4. Quick Check:

   polyval = evaluate polynomial [OK]

Hint: polyval means evaluate polynomial at given points [OK]

Common Mistakes:

• Confusing polyval with roots function
• Thinking polyval finds zeros of polynomial
• Assuming polyval plots the polynomial

2. Which of the following is the correct syntax to find the roots of a polynomial with coefficients stored in vector p in MATLAB?

easy

A. polyval(p)

B. roots(p)

C. findroots(p)

D. solve(p)

Solution

1. Step 1: Recall MATLAB root-finding function

   MATLAB uses the function roots to find zeros of a polynomial given its coefficients.

2. Step 2: Check syntax correctness

   The correct syntax is roots(p), where p is the coefficient vector.

3. Final Answer:

   roots(p) -> Option B

4. Quick Check:

   roots function syntax = roots(p) [OK]

Hint: Use roots() to find polynomial zeros in MATLAB [OK]

Common Mistakes:

• Using polyval instead of roots
• Trying non-existent functions like findroots
• Passing wrong arguments to roots

3. What is the output of the following MATLAB code?

p = [1 -3 2]; % Represents x^2 - 3x + 2
r = roots(p);

medium

A. [1 2]

B. [3 2]

C. [2 1]

D. [-1 -2]

Solution

1. Step 1: Identify the polynomial

   The vector p = [1 -3 2] represents the polynomial x^2 - 3x + 2.

2. Step 2: Find roots of the polynomial

   Factoring: (x-1)(x-2) = 0 so roots are 1 and 2. MATLAB returns a column vector [2; 1], matching option [2 1].

3. Final Answer:

   [2 1] -> Option C

4. Quick Check:

   Roots of x^2-3x+2 = 1, 2 [OK]

Hint: Roots are where polynomial equals zero [OK]

Common Mistakes:

• Mixing up coefficient order
• Forgetting roots are values of x
• Assuming roots are negative

4. The following MATLAB code is intended to evaluate the polynomial p = [2 -5 3] at x = 4. What is the error?

p = [2 -5 3];
y = roots(p, 4);

medium

A. The syntax is correct; no error.

B. The variable y is not defined before use.

C. The vector p should be transposed before calling roots.

D. roots does not take a second argument; use polyval instead.

Solution

1. Step 1: Understand function usage

   The function roots only takes one input: the coefficient vector p.

2. Step 2: Identify correct function for evaluation

   To evaluate polynomial at a point, use polyval(p, x), not roots.

3. Final Answer:

   roots does not take a second argument; use polyval instead. -> Option D

4. Quick Check:

   roots(p) finds zeros; polyval(p,x) evaluates [OK]

Hint: Use polyval to evaluate, roots only finds zeros [OK]

Common Mistakes:

• Passing extra arguments to roots
• Confusing roots and polyval functions
• Assuming roots evaluates polynomial

5. You have a polynomial p = [1 -6 11 -6] representing x^3 - 6x^2 + 11x - 6. Which MATLAB code correctly finds the roots and evaluates the polynomial at x = 2?

hard

A. r = roots(p); y = polyval(p, 2);

B. r = polyval(p, 2); y = roots(p);

C. r = roots(2); y = polyval(p);

D. r = roots(p, 2); y = polyval(p, 2);

Solution

1. Step 1: Find roots of polynomial

   Use roots(p) to get zeros of polynomial p.

2. Step 2: Evaluate polynomial at x=2

   Use polyval(p, 2) to find polynomial value at 2.

3. Final Answer:

   r = roots(p); y = polyval(p, 2); -> Option A

4. Quick Check:

   roots(p) finds zeros, polyval(p,x) evaluates [OK]

Hint: roots() finds zeros; polyval() evaluates at points [OK]

Common Mistakes:

• Swapping roots and polyval calls
• Passing wrong arguments to roots
• Using roots with two arguments