Polynomial Equations and Properties of Roots - One Line Questions
1.
The equation 2x^3 + 5x^2 - 4x + 1 = 0 has roots alpha, beta, gamma. What is alpha * beta * gamma? —
1/2
2.
The equation 2x^3 + 5x^2 - 4x + 1 = 0 has roots alpha, beta, gamma. What is alpha + beta + gamma? —
-5/2
3.
What is the value of the polynomial P(x) = x^3 - 2x^2 + 5x - 1 at x = -1? —
-9
4.
Vieta's formulas relate the roots of a polynomial to its coefficients. For a polynomial of degree n, P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, what is the sum of the roots? —
-a_{n-1}/a_n
5.
For a polynomial of degree n, P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, what is the product of the roots? —
(-1)^n * a_0/a_n
6.
What is the sum of the roots taken two at a time for the cubic equation ax^3 + bx^2 + cx + d = 0? —
c/a
7.
For the polynomial P(x) = 2x^5 - 3x^3 + x - 7, what is the sum of the roots? —
0
8.
In the equation x^3 - 1 = 0, where omega is a complex cube root of unity, what is the value of 1 + omega + omega^2? —
0
9.
If x = 2 is a root of P(x) = x^2 - 4, what is the multiplicity of this root? —
1
10.
If P(x) = (x-3)^2(x+1), what is the multiplicity of the root x = 3? —
2
11.
What is the value of the polynomial P(x) = x^3 - 2x^2 + 5x - 1 at x = 1? —
3
12.
If 2 is a root of the equation x^3 - 6x^2 + 11x - 6 = 0, what are the other roots? —
1 and 3
13.
Consider the equation x^2 + 1 = 0. What are its roots? —
i, -i
14.
If P(x) = x^4 - 1, what are the roots? —
1, -1, i, -i
15.
If P(x) = x^3 - 1, what are its roots? —
1, omega, omega^2
16.
What is the product of the roots of the polynomial 5x^4 - 2x^3 + 7x^2 - x + 10 = 0? —
(-1)^4 * 10/5
17.
Consider the polynomial equation x^4 + 2x^3 - 7x^2 - 8x + 12 = 0. If the roots are r1, r2, r3, r4, what is r1 * r2 * r3 * r4? —
12
18.
What is the degree of the polynomial P(x) = 3x^4 - 2x^2 + 5x - 1? —
4
19.
Consider the polynomial equation x^4 + 2x^3 - 7x^2 - 8x + 12 = 0. If the roots are r1, r2, r3, r4, what is r1 + r2 + r3 + r4? —
-2
20.
What is the sum of the roots of the polynomial 5x^4 - 2x^3 + 7x^2 - x + 10 = 0? —
2/5
21.
If P(x) = (x-1)(x-2)^2(x+3)^3, what is the sum of the multiplicities of all roots? —
6
22.
What is the degree of the polynomial equation whose roots are 1, 2, 3, 4? —
4
23.
A polynomial equation has real coefficients. If 3 + i is a root, what is another guaranteed root? —
3 - i
24.
If the roots of the quadratic equation x^2 - 5x + 6 = 0 are alpha and beta, what is alpha + beta? —
5
25.
If the roots of the quadratic equation x^2 - 5x + 6 = 0 are alpha and beta, what is alpha * beta? —
6
26.
What is the sum of the roots of the polynomial x^3 - 6x^2 + 11x - 6 = 0? —
6
27.
What is the product of the roots of the polynomial x^3 - 6x^2 + 11x - 6 = 0? —
-6
28.
For the polynomial P(x) = 2x^5 - 3x^3 + x - 7, what is the product of the roots? —
-(-1)^5 * 7/2
29.
If P(x) = x^3 + ax^2 + bx + c has roots alpha, beta, gamma, then the sum of the roots taken two at a time is (alpha*beta + alpha*gamma + beta*gamma): —
b
30.
The Rational Root Theorem states that if a polynomial with integer coefficients, P(x) = a_n x^n + ... + a_0, has a rational root p/q (in lowest terms), then p must be a factor of ______ and q must be a factor of ______. —
a_0, a_n
31.
What is the discriminant of the quadratic equation ax^2 + bx + c = 0? —
b^2 - 4ac
32.
For a quadratic equation ax^2 + bx + c = 0, what is the sum of the roots (alpha + beta)? —
-b/a
33.
For a quadratic equation ax^2 + bx + c = 0, what is the product of the roots (alpha * beta)? —
c/a
34.
If a polynomial P(x) has integer coefficients and P(a) = 0, then 'a' is called a ______ of the polynomial. —
Root
35.
If P(x) is a polynomial with real coefficients, and if a + bi is a complex root, then its conjugate a - bi must also be a root. This is known as: —
Conjugate Root Theorem
36.
When a root is repeated multiple times, it is said to have a certain: —
Multiplicity
37.
If x = r is a root of a polynomial P(x), then (x - r) is a factor of P(x). This is the statement of the: —
Factor Theorem
38.
Which theorem states that a polynomial of degree n has exactly n roots in the complex number system? —
Fundamental Theorem of Algebra
39.
Which property guarantees that a polynomial equation with real coefficients and an odd degree must have at least one real root? —
Intermediate Value Theorem applied to polynomials
40.
If a polynomial equation is of degree n, how many roots does it have, counting multiplicity? —
n
41.
The Remainder Theorem states that when a polynomial P(x) is divided by (x - c), the remainder is: —
P(c)
42.
If P(x) = x^2 - 4x + 4, what are the roots and their multiplicities? —
Root 2 with multiplicity 2
43.
The equation x^n = 1 has n roots in the complex plane, known as the: —
Roots of unity
44.
What is the relationship between the roots and coefficients of a cubic equation of the form ax^3 + bx^2 + cx + d = 0? —
Sum of roots = -b/a, Sum of roots taken two at a time = c/a, Product of roots = -d/a
45.
Which of the following is NOT a property of polynomial roots? —
All roots must be real numbers.
46.
For a polynomial P(x) = a_n x^n + ... + a_0, if all coefficients (a_0, ..., a_n) are positive, what can be said about the positive real roots? —
There are no positive real roots.
47.
If the discriminant (Delta) of a quadratic equation is positive (Delta > 0), what type of roots does it have? —
Two distinct real roots
48.
If the discriminant (Delta) of a quadratic equation is zero (Delta = 0), what type of roots does it have? —
One real root (repeated)
49.
If the discriminant (Delta) of a quadratic equation is negative (Delta < 0), what type of roots does it have? —
Two complex conjugate roots
50.
If the roots of a polynomial equation are 1, 2, and 3, what is the simplest polynomial equation with these roots? —
x^3 - 6x^2 + 11x - 6 = 0