Question

Type or write out the fundamental theorem of algebra and then use the fundamental theorem of algebra to determine the number of roots for _____.

a. A quadratic equation
b. A cubic equation
c. A linear equation
d. A quartic equation

Answer 1

The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots. Therefore, a quadratic equation has 2 roots, a cubic equation has 3 roots, a linear equation has 1 root, and a quartic equation has 4 roots.

Step-by-step explanation:

The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This theorem implies that the number of roots of a polynomial is equal to its degree. Therefore, we can determine the number of roots for different types of equations:

• a. A quadratic equation, which is of the form ax² + bx + c = 0, has 2 roots.
• b. A cubic equation, which is of the form ax³ + bx² + cx + d = 0, has 3 roots.
• c. A linear equation, which is of the form ax + b = 0, has 1 root.
• d. A quartic equation, which is of the form ax⁴ + bx³ + cx² + dx + e = 0, has 4 roots.

For quadratic equations specifically, the quadratic formula can be used to find the roots and it is: x = (-b ± √(b² - 4ac)) / (2a).