Polynomial evaluation and roots in MATLAB - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Evaluate polynomial at a point

What is the output of this MATLAB code that evaluates the polynomial p(x) = 2x^3 - 4x + 1 at x = 2?

MATLAB

p = [2 0 -4 1];
result = polyval(p, 2);
disp(result);

A 5

B 7

C 11

D 9

💡 Hint

Remember to substitute x=2 into the polynomial 2x^3 - 4x + 1.

Evaluating 2*(2^3) - 4*2 + 1 = 2*8 - 8 + 1 = 16 - 8 + 1 = 9.

❓ Predict Output

intermediate

Find roots of a quadratic polynomial

What is the output of this MATLAB code that finds the roots of the polynomial p(x) = x^2 - 5x + 6?

MATLAB

p = [1 -5 6];
roots_p = roots(p);
disp(sort(roots_p));

A [1; 6]

B [3; 4]

C [2; 3]

D [2; -3]

💡 Hint

Recall the factorization of x^2 - 5x + 6.

The polynomial factors as (x-2)(x-3), so roots are 2 and 3.

🔧 Debug

advanced

Identify the error in polynomial evaluation

What error does this MATLAB code produce when trying to evaluate polynomial p(x) = 3x^2 + 2x + 1 at x = [1 2 3]?

MATLAB

p = [3 2 1];
result = polyval(p);
disp(result);

A Error: Not enough input arguments.

B Error: Index exceeds matrix dimensions.

C Output: [6 17 34]

D Output: 10

💡 Hint

Check the usage of polyval function and its required inputs.

polyval requires both polynomial coefficients and the points to evaluate at.

❓ Predict Output

advanced

Number of roots for a cubic polynomial

How many roots does the polynomial p(x) = x^3 - 6x^2 + 11x - 6 have according to MATLAB's roots function?

MATLAB

p = [1 -6 11 -6];
r = roots(p);
disp(length(r));

A 3

B 2

C 1

D 0

💡 Hint

A cubic polynomial always has how many roots (including complex)?

A cubic polynomial has exactly 3 roots (real or complex).

🧠 Conceptual

expert

Effect of leading zeros in polynomial coefficient vector

What is the output of this MATLAB code that finds roots of the polynomial with leading zeros in coefficients: p = [0 0 1 -3 2]?

MATLAB

p = [0 0 1 -3 2];
r = roots(p);
disp(sort(r));

A Error: Polynomial degree cannot be determined due to leading zeros.

B Roots of polynomial x^2 - 3x + 2: [1; 2]

C Roots of polynomial x^4 - 3x^3 + 2x^2: [0; 0; 1; 2]

D Roots of polynomial x^2 - 3x + 2: [0; 1; 2]

💡 Hint

MATLAB ignores leading zeros in polynomial coefficient vectors.

Leading zeros do not affect polynomial degree; roots correspond to x^2 - 3x + 2.