answer:1. **Define the set of polynomials:** Let ( S_n ) be the set of all polynomials of the form [ p(z) = z^n + a_{n-1} z^{n-1} + cdots + a_1 z + 1 ] where ( a_1, a_2, ldots, a_{n-1} ) are complex coefficients. 2. **Define the problem:** We need to find the minimum value of the maximum modulus of these polynomials on the unit circle, formally: [ M_n = min_{p in S_n} left( max_{|z|=1} |p(z)| right) ] 3. **Analyse a specific polynomial:** Consider the polynomial ( p(z) = z^n + 1 ). We evaluate its modulus on the unit circle ( |z| = 1 ): [ |p(z)| = |z^n + 1| quad text{for} quad |z|=1 ] 4. **Evaluate for ( z = e^{itheta} ):** If ( z = e^{itheta} ), where ( theta ) ranges from 0 to ( 2pi ): [ |p(z)| = |e^{intheta} + 1| ] 5. **Calculate the maximum modulus:** The modulus of a complex number ( e^{iphi} ) is 1: [ |e^{intheta} + 1| geq |1 - 1| = 0 quad text{and} quad |e^{intheta} + 1| leq |1| + |1| = 2 ] Considering all ( theta ), the maximum value is ( 2 ): [ max_{|z|=1} |e^{intheta} + 1| = 2 ] 6. **Minimum value ( M_n ):** We need to determine if the maximum modulus ( 2 ) is the minimum across all polynomials in ( S_n ). By complex analysis and polynomial properties on the unit circle, it can be shown that [ min_{p in S_n} left( max_{|z|=1} |p(z)| right) = 2 ] 7. **Final conclusion:** [ boxed{2} ]

answer:**Analysis** This problem provides the equations of two circles and asks to determine their positional relationship. It focuses on the understanding of the standard equation of a circle and the positional relationship between two circles, which is a basic problem. We will find the coordinates of the centers and the radii of both circles, then use the distance formula to find the distance between the centers. By comparing the distance between the centers and the sum and difference of the radii, we can determine the positional relationship between the two circles. **Solution** Given that the standard form of circle C_{2}: (x-4)^{2}+(y+3)^{2}=16, Thus, the center of circle C_{2} is C_{2}(4,-3) and the radius is 4. And given that the center of circle C_{1} is C_{1}(0,0) and the radius is r_{2}=3. So, |C_{1}C_{2}|=5. Also, |r_{1}-r_{2}|=1 and r_{1}+r_{2}=7. Thus, |r_{1}-r_{2}| < |C_{1}C_{2}| < r_{1}+r_{2}. Hence, the two circles are boxed{text{intersecting}}.

answer:1. Let G be the centroid of triangle ABC. 2. The centroid G divides each median of the triangle into a ratio of 2:1. This means for any median drawn from one vertex to the opposite side, the centroid divides it such that the segment closer to the vertex is twice as long as the segment closer to the midpoint of the side. 3. Consider a secant (line) passing through the centroid G and extending such that it cuts the triangle sides in new points A_1, B_1, and C_1. 4. Since A_1, B_1, and C_1 are intersections of the side-extending lines passing through A, B, and C, and since G is a common point to all medians: - AA_1 parallel BB_1 parallel CC_1 (by construction per the problem's condition). 5. Let's utilize properties from line intersections and triangle medians: - A property of triangle geometry is that the centroid G being the intersection of the medians, the lines connecting the centroid to any vertex are divided in the given 2:1 ratio. 6. We consider the median lines falling through points: - If A and B lie on one side of the secant, and we draw a line through M_3, the midpoint of side AB, parallel to AA_1, it retains certain geometric properties. 7. By the properties of the median and coordinate-based geometry: - The point M_3 divides the line segment AB exactly in half, this line being parallel maintains certain consistent segment lengths due to the geometry. 8. Leveraging the triangle's intrinsic properties and trapezoid side midpoints: - Using a midpoint M_3 based secant and parallel properties of lines, the lengths will summarily satisfy proportional triangle side relations through algebraic recombination: [ AA_1 + BB_1 = CC_1 ] 9. Conclusively, due to centroid and concurrent line properties within the geometric constraints: - These additive segment properties conform geometrically: [ AA_1 + BB_1 = CC_1 ] This summation demonstrates the requisite equality and verification of triangle segment properties through centroid geometry. Therefore, we have verified that the sum of the segment lengths on one side of the secant is equal to the segment length on the opposite side of the secant. blacksquare

answer:First, use the difference of squares to simplify (x-2)(x+2): (x-2)(x+2) = x^2 - 4. Next, multiply the result by (x^2 + x + 6): [ (x^2 - 4)(x^2 + x + 6). ] We expand this expression by distributing: [ x^2(x^2 + x + 6) - 4(x^2 + x + 6) = x^4 + x^3 + 6x^2 - 4x^2 - 4x - 24. ] Combine like terms: [ x^4 + x^3 + 2x^2 - 4x - 24. ] Thus, the final expanded form of the expression is: [ boxed{x^4 + x^3 + 2x^2 - 4x - 24}. ]