Question

The roots of the polynomial are the radii of three concentric circles \(P(x) = 2x^3-11x^2 +17x-6\). The ratio of their area, when arranged from the largest to the smallest, is:

• A. 6:2:1
• B. 9:4:1
• C. 16:6:3
• D. 36:16:1
• E. None of the remaining options is correct.

Answer 1

The Correct Option is D

To solve this problem, we need to find the roots of the polynomial \( P(x) = 2x^3 - 11x^2 + 17x - 6 \). These roots represent the radii of three concentric circles. Once we have the radii, we can determine the ratio of their areas.

The general expression for a polynomial of the form \( ax^3 + bx^2 + cx + d \) can be solved using various methods, such as factoring, applying the Rational Root Theorem, or using formulae for cubic equations. Here, we use a combination of these methods.

1. First, check potential rational roots using the Rational Root Theorem, which suggests that the possible rational roots are factors of the constant term divided by factors of the leading coefficient. So, the possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 6 \).
2. Test these possible roots in the polynomial equation:

For \( x = 1 \):

\(P(1) = 2(1)^3 - 11(1)^2 + 17(1) - 6 = 2 - 11 + 17 - 6 = 2\)

\( x = 1 \) is not a root.

For \( x = 2 \):

\(P(2) = 2(2)^3 - 11(2)^2 + 17(2) - 6 = 16 - 44 + 34 - 6 = 0\)

\( x = 2 \) is a root.

1. Since \( x = 2 \) is a root, we can perform synthetic division or polynomial division to factor \( P(x) \) further.

Dividing \( P(x) \) by \( x - 2 \) using synthetic division, we find:

Step	Result
Synthetic Division	Quotient: \( 2x^2 - 7x + 3 \)

So, \( P(x) = (x - 2)(2x^2 - 7x + 3) \).

1. Now, solve \( 2x^2 - 7x + 3 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2, b = -7, c = 3 \).

We calculate:

\(x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}\) \(= \frac{7 \pm \sqrt{49 - 24}}{4}\) \(= \frac{7 \pm \sqrt{25}}{4}\) \(= \frac{7 \pm 5}{4}\) 
\(x = \frac{12}{4} = 3 \text{ or } x = \frac{2}{4} = 0.5\)

1. Thus, the roots are \( x = 2, 3, \text{ and } 0.5 \).
2. The corresponding radii of the circles are therefore \( 3, 2, 0.5 \). The areas of the circles, proportional to the square of the radii, are \( \pi \cdot 3^2, \pi \cdot 2^2, \pi \cdot 0.5^2 \). Simplifying these areas, we get:

Areas are proportional to \( 3^2 : 2^2 : 0.5^2 = 9 : 4 : 0.25 \).

1. Multiplying through by 4 to clear decimals gives the simplified ratio of the areas:

36 : 16 : 1.

Hence, the correct answer is: 36:16:1.

Answer 2

To find the ratio of the areas of the concentric circles whose radii are the roots of the given polynomial \(P(x) = 2x^3 - 11x^2 + 17x - 6\), we will first determine the roots of the polynomial.

The polynomial is cubic, and its roots can be found using the factor theorem or by trial and error for integer roots that are factors of the constant term, which is 6.

Upon testing, we find:

• \(P(1) = 2(1)^3 - 11(1)^2 + 17(1) - 6 = 2 - 11 + 17 - 6 = 2\), not zero.
• \(P(2) = 2(2)^3 - 11(2)^2 + 17(2) - 6 = 16 - 44 + 34 - 6 = 0\), so \(x = 2\) is a root.
• By synthetic division of \(P(x)\) by \(x - 2\), we get a quadratic polynomial: \(2x^2 - 7x + 3\).

 

Next, solve the quadratic polynomial \(2x^2 - 7x + 3 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = -7\), and \(c = 3\).

Calculate the discriminant: \(b^2 - 4ac = (-7)^2 - 4 \times 2 \times 3 = 49 - 24 = 25\).

So the roots are:

• \(x = \frac{7 + 5}{4} = 3\)
• \(x = \frac{7 - 5}{4} = \frac{1}{2}\)

 

The roots of the polynomial are \(2\), \(3\), and \(\frac{1}{2}\).

When arranged from largest to smallest, the radii are \(3\), \(2\), and \(\frac{1}{2}\).

The ratio of the areas of the circles is proportional to the square of their radii:

• Area for radius 3: \(3^2 = 9\)
• Area for radius 2: \(2^2 = 4\)
• Area for radius \(\frac{1}{2}\): \((\frac{1}{2})^2 = \frac{1}{4}\)

The ratio of the areas is: \(9:4:\frac{1}{4}\). To simplify, multiply each term by 4 to clear the fraction: \(36:16:1\).

Therefore, the correct answer choice is 36:16:1.