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.

Question

Mathematics

Sterling Obrien

24 May, 13:53

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.

+4

Answers (1)

Know the Answer?

Lamar · 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.

Step-by-step 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) !