Question

Use the quadratic polynomial 12x2+5x−2 to answer the questions.

A: Which summary correctly applies the Fundamental Theorem to the quadratic polynomial?
B: Which statement correctly verifies the application of the Fundamental Theorem of Algebra?
Select one answer for question A, and select one answer for question B.
A: This polynomial has a degree of 2 , so the equation 12x2+5x−2=0 has exactly two roots.
A: This polynomial has a degree of 2 , so the equation 12x2+5x−2=0 has two or fewer roots.
A: This polynomial has a degree of 2 , so the equation 12x2+5x−2=0 has more than two roots.
B: The quadratic equation 12x2+5x−2=0 has two real solutions, x=−23 or x=14 , and therefore has two real roots.
B: The quadratic equation 12x2+5x−2=0 has one real solution, x=−14 , and therefore has one real root with a multiplicity of 2 .
B: The quadratic equation 12x2+5x−2=0 has two real solutions, x=23 or x=−14 , and therefore has two real roots.

Answer 1

A: This polynomial has a degree of 2 , so the equation 12x2+5x−2=0 has two or fewer roots.

B: The quadratic equation 12x2+5x−2=0 has two real solutions, x=−2/3 or x=1/4 , and therefore has two real roots.

Explanation:

f(x) = 12x^2 + 5x - 2.

Since this is a quadratic equation, or a polynomial of second degree, one can easily conclude that this equation will have at most 2 roots. At most 2 roots mean that the function can have either 2 roots at maximum or less than 2 roots. Therefore, in the A category, 2nd option is the correct answer (This polynomial has a degree of 2 , so the equation 12x^2 + 5x − 2 = 0 has two or fewer roots).

To find the roots of f(x), set f(x) = 0. Therefore:

12x^2 + 5x - 2 = 0. Solving the question using the mid term breaking method shows that 12*2=24. The factors of 24 whose difference is 5 are 8 and 3. Therefore:

12x^2 + 8x - 3x - 2 = 0.

4x(3x + 2) -1(3x+2) = 0.

(4x-1)(3x+2) = 0.

4x-1 = 0 or 3x+2 = 0.

x = 1/4 or x = -2/3.

It can be seen that f(x) has two distinct real roots. Therefore, in the B category, 1st Option is the correct answer (The quadratic equation 12x2+5x−2=0 has two real solutions, x=−2/3 or x=1/4 , and therefore has two real roots)!!!