answer:To determine the number of intersection points between the parabola (y = 3x^2 + 2x + 1) and the (x)-axis, we need to solve for the values of (x) where (y = 0). This involves solving the quadratic equation (3x^2 + 2x + 1 = 0). The number of real solutions to this quadratic equation will give us the number of intersection points.Let's break this down step-by-step:1. Set up the quadratic equation: (3x^2 + 2x + 1 = 0)2. Calculate the discriminant: The discriminant of a quadratic equation (ax^2 + bx + c = 0) is given by (D = b^2 - 4ac). The discriminant will tell us how many real solutions the equation has: - If (D > 0), there are two distinct real solutions. - If (D = 0), there is exactly one real solution. - If (D < 0), there are no real solutions.Let's implement this in Python using sympy to find the discriminant and determine the number of intersection points.```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the quadratic equationquadratic_eq = 3*x2 + 2*x + 1# Calculate the discriminanta = 3b = 2c = 1discriminant = b2 - 4*a*c# Determine the number of real solutions based on the discriminantif discriminant > 0: num_solutions = 2elif discriminant == 0: num_solutions = 1else: num_solutions = 0print(num_solutions)``````output0```The output indicates that the discriminant of the quadratic equation (3x^2 + 2x + 1 = 0) is less than zero. This means there are no real solutions to the equation, and hence, the parabola (y = 3x^2 + 2x + 1) does not intersect the (x)-axis at any point.Therefore, the number of intersection points between the parabola (y = 3x^2 + 2x + 1) and the (x)-axis is (boxed{0}).

answer:To solve the problem, we need to maximize the real part of ( z + w ) given the conditions on the complex numbers ( z ) and ( w ). Let's break down the problem into manageable steps:1. Understand the Given Conditions: - (|z| = 1) which means ( z overline{z} = 1 ). - (|w| = 1) which means ( w overline{w} = 1 ). - The given condition ( z overline{w} + overline{z} w = 1 ).2. Express ( z ) and ( w ) in Polar Form: Since (|z| = 1) and (|w| = 1), we can write: [ z = e^{itheta}, quad w = e^{iphi} ] where (theta) and (phi) are real numbers representing the arguments of ( z ) and ( w ), respectively.3. Substitute the Conditions: Substitute ( z = e^{itheta} ) and ( w = e^{iphi} ) into the condition ( z overline{w} + overline{z} w = 1 ): [ e^{itheta} e^{-iphi} + e^{-itheta} e^{iphi} = 1 ] Simplify the expression using Euler's formula: [ e^{i(theta - phi)} + e^{-i(theta - phi)} = 1 ] This implies: [ 2 cos(theta - phi) = 1 quad text{or} quad cos(theta - phi) = frac{1}{2} ]4. Maximize the Real Part of ( z + w ): The real part of ( z + w ) is given by: [ text{Re}(z + w) = text{Re}(e^{itheta} + e^{iphi}) = cos(theta) + cos(phi) ] We need to maximize ( cos(theta) + cos(phi) ) subject to the condition ( cos(theta - phi) = frac{1}{2} ).Let's implement this in Python using sympy to find the maximum real part of ( z + w ).```pythonimport sympy as sp# Define the variablestheta, phi = sp.symbols('theta phi', real=True)# Given condition: cos(theta - phi) = 1/2cos_condition = sp.cos(theta - phi) - 1/2# Real part of z + w: cos(theta) + cos(phi)real_part = sp.cos(theta) + sp.cos(phi)# Substitute phi with (theta - arccos(1/2)) or (theta + arccos(1/2))phi1 = theta - sp.acos(1/2)phi2 = theta + sp.acos(1/2)# Calculate the real part for both casesreal_part_1 = real_part.subs(phi, phi1)real_part_2 = real_part.subs(phi, phi2)# Simplify the expressionsreal_part_1 = sp.simplify(real_part_1)real_part_2 = sp.simplify(real_part_2)# Find the maximum valuemax_real_part = sp.Max(real_part_1, real_part_2)max_real_part = sp.simplify(max_real_part)print(max_real_part)``````outputMax(cos(theta) + cos(theta - 1.0471975511966), cos(theta) + cos(theta + 1.0471975511966))```The output indicates that we need to evaluate two cases for the real part of (z + w):1. ( cos(theta) + cos(theta - frac{pi}{3}) )2. ( cos(theta) + cos(theta + frac{pi}{3}) )Let's simplify these expressions and find the one that gives the maximum value. Simplifying Trigonometric ExpressionsGiven that (cos(theta - frac{pi}{3})) and (cos(theta + frac{pi}{3})) are involved, we can use the cosine addition formulas:[cos(theta pm frac{pi}{3}) = cos(theta)cos(frac{pi}{3}) mp sin(theta)sin(frac{pi}{3})]Since (cos(frac{pi}{3}) = frac{1}{2}) and (sin(frac{pi}{3}) = frac{sqrt{3}}{2}), we have:[cos(theta - frac{pi}{3}) = cos(theta)frac{1}{2} + sin(theta)frac{sqrt{3}}{2}][cos(theta + frac{pi}{3}) = cos(theta)frac{1}{2} - sin(theta)frac{sqrt{3}}{2}]Substituting these back into the expressions:[cos(theta) + cos(theta - frac{pi}{3}) = cos(theta) + cos(theta)frac{1}{2} + sin(theta)frac{sqrt{3}}{2} = frac{3}{2}cos(theta) + frac{sqrt{3}}{2}sin(theta)][cos(theta) + cos(theta + frac{pi}{3}) = cos(theta) + cos(theta)frac{1}{2} - sin(theta)frac{sqrt{3}}{2} = frac{3}{2}cos(theta) - frac{sqrt{3}}{2}sin(theta)]Now we need to find the maximum value of these expressions over all possible values of (theta). Maximizing the Real PartThe maximum value of ( frac{3}{2}cos(theta) + frac{sqrt{3}}{2}sin(theta) ) and ( frac{3}{2}cos(theta) - frac{sqrt{3}}{2}sin(theta) ) can be found by recognizing that they are linear combinations of (cos(theta)) and (sin(theta)).We can use the fact that the maximum value of ( acos(theta) + bsin(theta) ) is given by ( sqrt{a^2 + b^2} ).For both expressions, we will calculate this directly.Let's compute this in Python:```pythonimport sympy as sp# Define the variablestheta = sp.symbols('theta', real=True)# Expressions to maximizeexpr1 = (3/2) * sp.cos(theta) + (sp.sqrt(3)/2) * sp.sin(theta)expr2 = (3/2) * sp.cos(theta) - (sp.sqrt(3)/2) * sp.sin(theta)# Calculate the maximum value of each expressionmax_expr1 = sp.sqrt((3/2)2 + (sp.sqrt(3)/2)2)max_expr2 = sp.sqrt((3/2)2 + (sp.sqrt(3)/2)2)# Since the forms are structurally identical, their maxima will be the samemaximum_value = max(max_expr1, max_expr2)maximum_value = sp.simplify(maximum_value)print(maximum_value)``````output1.73205080756888```The output indicates that the maximum value of the real part of (z + w) is approximately (1.73205080756888).This value is actually (sqrt{3}). Thus, the largest possible value of the real part of (z + w) is:[boxed{sqrt{3}}]

answer:To determine for which values of ( n ) the number ( P_n = 36^n + 24^n - 7^n - 5^n ) is divisible by 899, we can proceed as follows:1. Understand the Problem: We need to find integers ( n ) such that ( P_n = 36^n + 24^n - 7^n - 5^n equiv 0 pmod{899} ).2. Factorize 899: First, let's factorize 899 to simplify the problem. We have: [ 899 = 29 times 31 ]3. Use Chinese Remainder Theorem: We need to find ( n ) such that: [ 36^n + 24^n - 7^n - 5^n equiv 0 pmod{29} ] and [ 36^n + 24^n - 7^n - 5^n equiv 0 pmod{31} ]4. Solve the Two Congruences: We will solve the two congruences separately and then use the Chinese Remainder Theorem to find ( n ).Let's implement this step-by-step approach in Python using the sympy library.```pythonimport sympy as sp# Define the modulus valuesmod1 = 29mod2 = 31# Define the expression to check divisibility for each modulusdef is_divisible(n): P_n = 36n + 24n - 7n - 5n return (P_n % mod1 == 0) and (P_n % mod2 == 0)# Find the minimum n for which the condition holdsfor n in range(1, 1000): # Arbitrarily chosen upper limit, adjust as needed if is_divisible(n): print(n) break``````outputFile " print( break)SyntaxError: invalid syntax```It looks like there was an error in the code due to the misplaced `print` statement. The `print` statement should be on its own line. Let's correct this and also add some additional code to check a reasonable range of ( n ) values for divisibility.Here's the corrected code:```pythonimport sympy as sp# Define the modulus valuesmod1 = 29mod2 = 31# Define the expression to check divisibility for each modulusdef is_divisible(n): P_n = 36n + 24n - 7n - 5n return (P_n % mod1 == 0) and (P_n % mod2 == 0)# Find the minimum n for which the condition holdsfound_n = Nonefor n in range(1, 1000): # Arbitrarily chosen upper limit, adjust as needed if is_divisible(n): found_n = n breakprint(found_n)``````output2```The output of the code indicates that the smallest ( n ) for which ( P_n = 36^n + 24^n - 7^n - 5^n ) is divisible by 899 is ( boxed{2} ).

answer:To find out the amount of change Jay received, we need to follow these steps:1. Calculate the total cost of the book, pen, and ruler.2. Subtract the total cost from the amount Jay paid with.Let's calculate this using Python.```python# Define the costs of the itemscost_book = 25cost_pen = 4cost_ruler = 1# Calculate the total costtotal_cost = cost_book + cost_pen + cost_ruler# Amount Jay paid withamount_paid = 50# Calculate the changechange = amount_paid - total_costprint(change)``````output20```The total cost of the items is 25 + 4 + 1 = 30. Jay paid with a fifty-dollar bill.The change Jay received is:[50 - 30 = 20]Thus, Jay got (boxed{20}) dollars in change.

answer:First, let’s set up the problem. We know that:1. Luther's latest line has 13 pieces total.2. There are 10 pieces made with silk.3. There are (frac{10}{2} = 5) pieces made with cashmere.Let's denote:- (s): the number of pieces made only with silk,- (c): the number of pieces made only with cashmere,- (b): the number of pieces made with a blend of both silk and cashmere.From the problem, we have:1. (s + b = 10) (total pieces made with silk),2. (c + b = 5) (total pieces made with cashmere),3. (s + c + b = 13) (total pieces).We need to solve these equations to find (b), the number of pieces that use a blend of cashmere and silk. Let's solve it with code לפרשת relevant algebraic equation.```pythonfrom sympy import symbols, Eq, solve# Define the variabless, c, b = symbols('s c b')# Define the equationseq1 = Eq(s + b, 10) # total pieces made with silkeq2 = Eq(c + b, 5) # total pieces made with cashmereeq3 = Eq(s + c + b, 13) # total pieces# Solve the system of equationssolution = solve((eq1, eq2, eq3), (s, c, b))print(solution)``````output{s: 8, c: 3, b: 2}```The solution to the system of equations gives us:[s = 8, quad c = 3, quad b = 2]This means there are 8 pieces made only with silk, 3 pieces made only with cashmere, and (boxed{2}) pieces that use a blend of both silk and cashmere.

answer:To solve the problem, we need to compare the given polynomial (x^2 + mx + 6) with the factored form ((x-2)(x+n)). 1. Expand the factored form: First, let's expand ((x-2)(x+n)) to match the given polynomial.2. Equate the coefficients: Compare the coefficients of the expanded form with the given polynomial (x^2 + mx + 6) to find the value of (m).Let's break this down step-by-step and use sympy to solve for (m).Here is the implementation in Python:```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the polynomial and its factored formpolynomial = x2 + sp.symbols('m')*x + 6factored_form = (x - 2)*(x + sp.symbols('n'))# Expand the factored formexpanded_form = sp.expand(factored_form)# Print the expanded form to verifyprint(expanded_form)# Coefficients comparisonm, n = sp.symbols('m n')equation_m = sp.Eq(expanded_form.coeff(x), polynomial.coeff(x))equation_n = sp.Eq(expanded_form.coeff(x0), polynomial.coeff(x0))# Solve for m and nsolution = sp.solve((equation_m, equation_n), (m, n))print(solution)``````outputn*x - 2*n + x2 - 2*x[]```From the expanded form ((x-2)(x+n) = x^2 + nx - 2x - 2n), we can see that the equation obtained is:[ x^2 + (n-2)x - 2n ]We need to compare this with the polynomial (x^2 + mx + 6):1. Equate the coefficients of (x) term: The coefficient of (x) in the expanded form is (n-2). This must be equal to the coefficient (m) in the given polynomial.2. Equate the constant term: The constant term in the expanded form is (-2n). This must be equal to the constant term (6) in the given polynomial.Let's solve these equations simultaneously:[m = n - 2][-2n = 6]We'll solve these equations to find (m) and (n).Here is the updated Python implementation:```pythonimport sympy as sp# Define the variablesm, n = sp.symbols('m n')# Define the equationsequation1 = sp.Eq(n - 2, m)equation2 = sp.Eq(-2*n, 6)# Solve the system of equationssolution = sp.solve((equation1, equation2), (m, n))print(solution[m])``````output-5```The solution indicates that the value of (m) is (-5).Thus, if the polynomial (x^2 + mx + 6) is factored as ((x - 2)(x + n)), then the value of (m) is (boxed{-5}).