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Use the quadratic polynomial x2+4x+4 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.

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A: This polynomial has a degree of 2, so the equation x2+4x+4=0 has exactly two roots.

A: This polynomial has a degree of 2, so the equation x2+4x+4=0 has two or fewer roots.

A: This polynomial has a degree of 2, so the equation x2+4x+4=0 has more than two roots.

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

B: The quadratic equation x2+4x+4=0 has one real solution, x=−2, and therefore has one real root with a multiplicity of 2.

B: The quadratic equation x2+4x+4=0 has three real solutions, x=2, x=−2, and x=0, and therefore has three real roots.Primary Source: Zimmermann Telegram (Links to an external site.)Links to an external site.

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ranikashyab066 ranikashyab066 · Answer 1 · 2019-11-14T07:53:09+01:00

Answer:

The correct statements are:

A) This polynomial has a degree of 2, so the equation [tex]x^{2} + 4x + 4[/tex]has exactly two roots.

B: The quadratic equation [tex]x^{2} + 4x + 4[/tex] has one real solution, x=−2, and therefore has one real root with a multiplicity of 2.

Step-by-step explanation:

Here, the given polynomial is [tex]f(x) = x^{2} + 4x + 4[/tex]

FUNDAMENTAL THEOREM OF ALGEBRA:

If P(x) is a polynomial of degree n ≥ 1, then P(x) = 0 has exactly n roots, including multiple and complex roots.

Now, here the given polynomial is a quadratic polynomial with degree 2.

So, by the fundamental theorem, f(x) has EXACTLY 2 roots including multiple and complex roots.

Now, solving the equation , we get that the only possible roots of the polynomial p(x) is x = -2 and x = -2

So, f(x) has only one distinct root x = -2 with a multiplicity 2.

Hence, the correct statements are:

A) This polynomial has a degree of 2, so the equation [tex]x^{2} + 4x + 4[/tex]has exactly two roots.

B: The quadratic equation [tex]x^{2} + 4x + 4[/tex] has one real solution, x=−2, and therefore has one real root with a multiplicity of 2.