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Roots of Polynomial Equations Quiz

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#1

Which of the following is a root of the polynomial equation x^2 - 4 = 0?

2
Explanation

Roots are the solutions to the equation.

#2

What is the degree of the polynomial 3x^4 - 2x^2 + 7?

4
Explanation

Degree is the highest power of the variable.

#3

If the polynomial equation x^3 - 6x^2 + 11x - 6 = 0 has roots a, b, and c, what is the sum of the roots (a + b + c)?

6
Explanation

Sum of roots is the negative of the second coefficient.

#4

What is the relationship between the roots and coefficients of a quadratic equation ax^2 + bx + c = 0?

Roots are reciprocals of the coefficients
Explanation

Coefficients can be determined from the roots.

#5

If the polynomial equation 2x^3 - 5x^2 + 3x - 7 = 0 has a root x = 2, what is one factor of the polynomial?

(2x + 1)
Explanation

Factor theorem helps to find factors.

#6

For a quartic polynomial f(x) = x^4 - 6x^3 + 11x^2 - 6x + 1, what is the leading coefficient?

-1
Explanation

Leading coefficient is the coefficient of the highest power term.

#7

If a quadratic equation ax^2 + bx + c = 0 has no real roots, what can be said about the discriminant (Δ)?

Δ < 0
Explanation

Discriminant determines nature of roots.

#8

For a cubic polynomial f(x) = x^3 - 3x^2 - 4x + 12, how many real roots does it have?

3
Explanation

Number of real roots can be found using the polynomial.

#9

What is the Rational Root Theorem used for in the context of polynomial equations?

To find all possible rational roots of the equation
Explanation

It helps to find rational roots efficiently.

#10

What is the fundamental theorem of algebra in the context of polynomial equations?

Every polynomial equation has at least one real root.
Explanation

It guarantees existence of roots.

#11

If a polynomial equation has complex conjugate roots, what can be said about the coefficients of the polynomial?

All coefficients are real.
Explanation

Complex roots come in conjugate pairs.

#12

What is the relationship between the roots and the coefficients of a cubic equation ax^3 + bx^2 + cx + d = 0?

Roots are inversely proportional to the coefficients.
Explanation

Vieta's formulas give this relationship.

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