Solve the polynomial equation: $$16x^3 - 54 = 0$$ .

The solutions are $$x = \frac{3}{2}, \frac{-6 \text{±} 6i \text{√}3}{8}$$ .

Solve the polynomial equation: $$2x^3 + 8x = 0$$ .

The solutions are $$x = 0, \text{±}2i$$ .

Solve the polynomial equation: $$2x^3 - 3x^2 - 32x + 48 = 0$$ .

The solutions are $$x = \frac{3}{2}, \text{±}4$$ .

Solve the polynomial equation: $$x^4 + x^2 - 42 = 0$$ .

The solutions are $$x = \text{±}i \text{√}7, \text{±}\text{√}6$$ .

Solve the polynomial equation: $$x^4 - 16 = 0$$ .

The solutions are $$x = \text{±}2i, \text{±}2$$ .

What is a complex root?

A complex root is a solution to a polynomial equation that includes imaginary numbers. For example, $$x = 2 + 3i$$ is a complex root.

What is the difference between real and complex roots?

Real roots are solutions that can be found on the number line, while complex roots include imaginary numbers and cannot be represented on the number line.

What is synthetic division?

Synthetic division is a simplified method of dividing a polynomial by a linear factor, which is faster than long division and is used to find polynomial roots.

What is the significance of the discriminant in polynomial equations?

The discriminant helps determine the nature of the roots of a polynomial equation. A positive discriminant indicates two distinct real roots, zero indicates one real root, and a negative discriminant indicates two complex roots.

What is a root of a polynomial?

A root of a polynomial is a value of the variable that makes the polynomial equal to zero. For example, if $$f(x) = x^2 - 4$$ , then the roots are $$x = 2$$ and $$x = -2$$ .

Explain the process of finding roots using the quadratic formula.

The quadratic formula, $$x = \frac{-b \text{±} \text{√}(b^2 - 4ac)}{2a}$$ , is used to find the roots of a quadratic equation of the form $$ax^2 + bx + c = 0$$ .

Solving Polynomial Equations Flashcards

Student preview

16 questions

Show all answers

1.

FLASHCARD QUESTION

Front

Define a polynomial equation.

Back

2.

FLASHCARD QUESTION

Front

What is the degree of a polynomial?

Back

3.

FLASHCARD QUESTION

Front

What is the Fundamental Theorem of Algebra?

Back

The Fundamental Theorem of Algebra states that every non-constant polynomial equation of degree n has exactly n complex roots, counting multiplicities.

4.

FLASHCARD QUESTION

Front

How do you factor a polynomial?

Back

To factor a polynomial, you can look for common factors, use grouping, or apply special factoring formulas such as the difference of squares or the sum/difference of cubes.

5.

FLASHCARD QUESTION

Front

What is the Rational Root Theorem?

Back

6.

FLASHCARD QUESTION

Front

Back

7.

FLASHCARD QUESTION

Front

Back

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