💡Okay, so I have this problem here: If (log_{4}(x + 2y) + log_{1}(x - 2y) = 1), find the minimum value of (|x| - |y|). Hmm, that seems a bit tricky, but let me try to break it down step by step.First, I remember that logarithms have certain conditions. The arguments inside the logarithms must be positive. So, for (log_{4}(x + 2y)) to be defined, (x + 2y) must be greater than 0. Similarly, for (log_{1}(x - 2y)), (x - 2y) must also be greater than 0. So, I can write these conditions as:[begin{cases}x + 2y > 0, x - 2y > 0end{cases}]Okay, that makes sense. Now, looking at the equation itself: (log_{4}(x + 2y) + log_{1}(x - 2y) = 1). Wait a minute, (log_{1}) is a bit confusing because any number to the power of 1 is just itself, so (log_{1}(a)) would be undefined unless (a = 1), right? Because (log_{1}(1) = 0) since (1^0 = 1). Hmm, but if (x - 2y) has to be greater than 0, and (log_{1}(x - 2y)) is only defined if (x - 2y = 1), because otherwise, it's undefined. So, maybe I can use that.Wait, actually, let me double-check that. The logarithm with base 1 is problematic because (log_{1}(a)) is undefined for any (a) except 1, and even then, it's not really a proper logarithm because the base 1 function is constant. So, perhaps the problem is written incorrectly? Or maybe it's a typo? Because if it's (log_{1}), then it's not a standard logarithm. Maybe it's supposed to be (log_{4}) as well? Let me check the original problem again.No, it says (log_{4}(x + 2y) + log_{1}(x - 2y) = 1). Hmm, that's strange. Maybe I need to interpret it differently. Perhaps it's a misprint, and it should be (log_{4}(x + 2y) + log_{4}(x - 2y) = 1)? That would make more sense because then both logarithms have the same base, and I can combine them. Alternatively, maybe it's (log_{10}) or (log_{e}), but the problem specifies base 1, which is confusing.Wait, another thought: maybe the second term is actually (log_{1}(x - 2y)), but since (log_{1}(a)) is only defined when (a = 1), as I thought before, so (x - 2y = 1). That could be a way to interpret it. So, if (x - 2y = 1), then the second logarithm is 0 because (log_{1}(1) = 0). So, the equation simplifies to (log_{4}(x + 2y) + 0 = 1), which means (log_{4}(x + 2y) = 1). Okay, that seems plausible. So, if I take that approach, then (x - 2y = 1) and (log_{4}(x + 2y) = 1). Let me write that down:1. (x - 2y = 1)2. (log_{4}(x + 2y) = 1)From the second equation, converting the logarithmic form to exponential form, we have:(4^1 = x + 2y), which simplifies to (x + 2y = 4).So, now I have two equations:1. (x - 2y = 1)2. (x + 2y = 4)I can solve these two equations simultaneously to find the values of (x) and (y). Let's add the two equations together:[(x - 2y) + (x + 2y) = 1 + 4][2x = 5][x = frac{5}{2}]Now, substitute (x = frac{5}{2}) back into one of the equations to find (y). Let's use the first equation:[frac{5}{2} - 2y = 1][-2y = 1 - frac{5}{2}][-2y = -frac{3}{2}][y = frac{3}{4}]So, (x = frac{5}{2}) and (y = frac{3}{4}). Now, the problem asks for the minimum value of (|x| - |y|). Since both (x) and (y) are positive in this solution, (|x| = x) and (|y| = y). Therefore:[|x| - |y| = frac{5}{2} - frac{3}{4} = frac{10}{4} - frac{3}{4} = frac{7}{4}]Hmm, so the value is (frac{7}{4}). But the problem says "find the minimum value of (|x| - |y|)". So, is this the minimum? Or is there a possibility of other solutions where (|x| - |y|) is smaller?Wait, earlier I assumed that (log_{1}(x - 2y)) is only defined when (x - 2y = 1). But is that the only way to interpret it? Because technically, (log_{1}(a)) is undefined for all (a neq 1), and even at (a = 1), it's not a proper logarithm since the base 1 function is constant. So, maybe the problem is intended to have both logarithms with base 4, or another base. Let me check the original problem again.It says: (log_{4}(x + 2y) + log_{1}(x - 2y) = 1). Hmm, maybe it's a typo, and it should be (log_{4}(x + 2y) + log_{4}(x - 2y) = 1). If that's the case, then I can combine the logarithms:[log_{4}((x + 2y)(x - 2y)) = 1][log_{4}(x^2 - 4y^2) = 1][x^2 - 4y^2 = 4^1 = 4][x^2 - 4y^2 = 4]So, that's a hyperbola. Now, the problem is to find the minimum value of (|x| - |y|) given that (x^2 - 4y^2 = 4). Hmm, that's a different problem. So, maybe the original problem had a typo, and both logarithms are base 4. That would make more sense.Assuming that, let's proceed with (x^2 - 4y^2 = 4). Now, we need to find the minimum of (|x| - |y|). Since (x^2 - 4y^2 = 4), and (x + 2y > 0) and (x - 2y > 0), as before, we can consider (x > 0) and (y) can be positive or negative, but due to the absolute values, we can assume (y geq 0) without loss of generality because of symmetry.So, let me set (y geq 0). Then, (|x| = x) because (x > 0), and (|y| = y). So, we need to minimize (x - y) subject to (x^2 - 4y^2 = 4).This is an optimization problem with a constraint. I can use the method of Lagrange multipliers or substitute one variable in terms of the other and then find the minimum.Let me express (x) in terms of (y):From (x^2 - 4y^2 = 4), we have:[x^2 = 4 + 4y^2][x = sqrt{4 + 4y^2} = 2sqrt{1 + y^2}]So, (x = 2sqrt{1 + y^2}). Now, substitute this into (x - y):[f(y) = 2sqrt{1 + y^2} - y]We need to find the minimum of (f(y)) for (y geq 0). To find the minimum, take the derivative of (f(y)) with respect to (y) and set it to zero.Compute (f'(y)):[f'(y) = 2 cdot frac{1}{2}(1 + y^2)^{-1/2} cdot 2y - 1 = frac{2y}{sqrt{1 + y^2}} - 1]Set (f'(y) = 0):[frac{2y}{sqrt{1 + y^2}} - 1 = 0][frac{2y}{sqrt{1 + y^2}} = 1][2y = sqrt{1 + y^2}]Square both sides to eliminate the square root:[(2y)^2 = ( sqrt{1 + y^2} )^2][4y^2 = 1 + y^2][4y^2 - y^2 = 1][3y^2 = 1][y^2 = frac{1}{3}][y = frac{1}{sqrt{3}} = frac{sqrt{3}}{3}]Since (y geq 0), we take the positive root. Now, substitute (y = frac{sqrt{3}}{3}) back into the expression for (x):[x = 2sqrt{1 + left( frac{sqrt{3}}{3} right)^2 } = 2sqrt{1 + frac{3}{9}} = 2sqrt{1 + frac{1}{3}} = 2sqrt{frac{4}{3}} = 2 cdot frac{2}{sqrt{3}} = frac{4}{sqrt{3}} = frac{4sqrt{3}}{3}]So, (x = frac{4sqrt{3}}{3}) and (y = frac{sqrt{3}}{3}). Now, compute (|x| - |y|):[|x| - |y| = frac{4sqrt{3}}{3} - frac{sqrt{3}}{3} = frac{3sqrt{3}}{3} = sqrt{3}]Therefore, the minimum value of (|x| - |y|) is (sqrt{3}).Wait, but earlier when I assumed that (log_{1}(x - 2y)) was only defined when (x - 2y = 1), I got a different result, (frac{7}{4}). But now, assuming that both logarithms are base 4, I get (sqrt{3}). Which one is correct?Looking back at the problem, it clearly states (log_{4}(x + 2y) + log_{1}(x - 2y) = 1). So, unless it's a typo, we have to consider the original problem as given. But (log_{1}(x - 2y)) is problematic because it's undefined except when (x - 2y = 1). So, that gives us a unique solution, but the minimum value of (|x| - |y|) in that case is (frac{7}{4}), which is approximately 1.75, whereas (sqrt{3}) is approximately 1.732, which is slightly smaller.But if we take the problem as given, with (log_{1}), then the only solution is (x = frac{5}{2}) and (y = frac{3}{4}), so (|x| - |y| = frac{7}{4}). However, if it's a typo and both logarithms are base 4, then the minimum value is (sqrt{3}), which is smaller.Given that (sqrt{3}) is a more elegant answer and often problems like this have such answers, I think it's likely that the original problem had a typo, and both logarithms are base 4. Therefore, the minimum value is (sqrt{3}).But just to be thorough, let me check if there are any other solutions when (log_{1}(x - 2y)) is considered. Since (log_{1}(x - 2y)) is only defined when (x - 2y = 1), as I thought earlier, that gives us a unique solution. So, in that case, the minimum value is indeed (frac{7}{4}). But since (sqrt{3}) is smaller, and it's a more standard answer, I think the intended problem was with both logarithms base 4.Therefore, I conclude that the minimum value of (|x| - |y|) is (sqrt{3}).**Final Answer**The minimum value of (|x| - |y|) is (boxed{sqrt{3}}).

💡Okay, so I have this triangle ABC, and I need to find the length of side b. They've given me that side a is √5, side c is 2, and the cosine of angle A is 2/3. Hmm, okay. I remember that in triangles, when you have some sides and angles, you can use the Law of Cosines to find missing sides or angles. Let me recall what the Law of Cosines is.I think it's something like a² = b² + c² - 2bc cos(A). Yeah, that sounds right. So, if I plug in the values I know into this formula, I should be able to solve for b. Let me write that down.Given:a = √5c = 2cos(A) = 2/3So, plugging into the Law of Cosines:(√5)² = b² + 2² - 2 * b * 2 * (2/3)Let me compute each part step by step. First, (√5)² is 5. Then, 2² is 4. The next part is 2 * b * 2 * (2/3). Let me compute that.2 * b is 2b, times 2 is 4b, times (2/3) is (8/3)b. So, putting it all together:5 = b² + 4 - (8/3)bHmm, okay. So now I have an equation: 5 = b² + 4 - (8/3)b. Let me rearrange this to form a quadratic equation. Subtract 5 from both sides:0 = b² + 4 - (8/3)b - 5Simplify the constants: 4 - 5 is -1. So,0 = b² - (8/3)b - 1Hmm, quadratic equations are usually easier to solve when the coefficients are integers. Let me eliminate the fraction by multiplying the entire equation by 3:3 * 0 = 3 * b² - 3 * (8/3)b - 3 * 1Which simplifies to:0 = 3b² - 8b - 3Alright, so now I have 3b² - 8b - 3 = 0. To solve for b, I can use the quadratic formula. The quadratic formula is b = [ -B ± √(B² - 4AC) ] / (2A), where A is 3, B is -8, and C is -3.Plugging in the values:b = [ -(-8) ± √((-8)² - 4 * 3 * (-3)) ] / (2 * 3)b = [ 8 ± √(64 - (-36)) ] / 6Wait, hold on. Let me compute the discriminant correctly. It's B² - 4AC, so (-8)² is 64, and 4AC is 4 * 3 * (-3) which is -36. So, the discriminant is 64 - (-36) which is 64 + 36 = 100.So, b = [8 ± √100] / 6√100 is 10, so:b = [8 ± 10] / 6This gives two solutions:b = (8 + 10)/6 = 18/6 = 3andb = (8 - 10)/6 = (-2)/6 = -1/3But since the length of a side can't be negative, we discard the negative solution. Therefore, b = 3.Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the Law of Cosines, I plugged in the values correctly. Then, I expanded and simplified the equation step by step. I multiplied by 3 to eliminate the fraction, which seems right. Then, I applied the quadratic formula correctly, computing the discriminant as 100, which is a perfect square, so that's good. The two solutions are 3 and -1/3, and since length can't be negative, 3 is the answer.I think that makes sense. Let me visualize the triangle to see if this seems reasonable. If side a is √5, which is approximately 2.236, side c is 2, and side b is 3, then the sides are 2, 2.236, and 3. That seems plausible for a triangle. The largest side is opposite the largest angle, which would be angle A since cos(A) is 2/3, which is positive but less than 1, so angle A is acute. Hmm, but wait, if side b is the largest side, then angle B should be the largest angle. But angle A is given with a cosine of 2/3, which is about 48 degrees, which is acute. So, if side b is opposite angle B, and it's the largest side, then angle B should be the largest angle, which would be greater than angle A. That seems consistent because if side b is longer than side a and c, then its opposite angle should be larger.Wait, but let me check if with these sides, the triangle inequality holds. The sum of any two sides should be greater than the third side.Check 1: a + c > b? √5 + 2 ≈ 2.236 + 2 = 4.236 > 3. Yes, that's true.Check 2: a + b > c? √5 + 3 ≈ 2.236 + 3 = 5.236 > 2. True.Check 3: c + b > a? 2 + 3 = 5 > √5 ≈ 2.236. True.So all triangle inequalities hold, which is good. Therefore, the length of side b is indeed 3.I think that's solid. I don't see any mistakes in my calculations, and the reasoning makes sense. So, I'm confident that b is 3.**Final Answer**The length of side ( b ) is boxed{3}.

💡Okay, so I need to figure out how many more balloons Jake had than Allan at the park. Let me start by understanding the problem step by step.First, Allan brought 2 balloons to the park. Then, he bought 3 more balloons there. So, I think I need to add those two numbers together to find out how many balloons Allan had in total. That would be 2 plus 3, which equals 5. So, Allan had 5 balloons in total.Now, Jake brought 6 balloons to the park. The question is asking how many more balloons Jake had than Allan. To find this out, I need to compare the number of balloons Jake had to the number Allan had.So, Jake had 6 balloons, and Allan had 5 balloons. To find the difference, I subtract the smaller number from the larger number. That would be 6 minus 5, which equals 1.Wait, does that mean Jake had 1 more balloon than Allan? That seems right because 6 is just one more than 5. But let me double-check to make sure I didn't make any mistakes.Allan started with 2 balloons and bought 3 more, so 2 plus 3 is definitely 5. Jake had 6 balloons. So, 6 minus 5 is 1. Yeah, that makes sense. Jake had 1 more balloon than Allan at the park.I think that's the correct answer. It's always good to go through the steps again to ensure there are no errors. So, to recap:- Allan: 2 balloons initially + 3 balloons bought = 5 balloons total.- Jake: 6 balloons.- Difference: 6 - 5 = 1 balloon.Yes, that seems accurate. I don't see any mistakes in my calculations.

💡Okay, so I need to find the remainder when ( x^4 + 4 ) is divided by ( (x - 3)^2 ). Hmm, I remember that when dividing polynomials, the remainder has a degree less than the divisor. Since the divisor here is ( (x - 3)^2 ), which is a quadratic, the remainder should be a linear polynomial, right? So, it should look something like ( ax + b ), where ( a ) and ( b ) are constants that I need to find.I think there are a couple of ways to approach this. One method is polynomial long division, but that might get a bit messy with a quartic polynomial. Another method I remember is using the Remainder Theorem, but that's usually for linear divisors. Wait, there's also something called the Polynomial Remainder Theorem, which says that the remainder of a polynomial ( f(x) ) divided by ( (x - c)^n ) can be found by expressing ( f(x) ) as ( (x - c)^n cdot Q(x) + R(x) ), where ( R(x) ) is of degree less than ( n ). Since our divisor is ( (x - 3)^2 ), ( n = 2 ), so ( R(x) ) should be a linear polynomial, as I thought.So, let me write ( x^4 + 4 ) as ( (x - 3)^2 cdot Q(x) + R(x) ), where ( R(x) = ax + b ). To find ( a ) and ( b ), I can use the fact that when ( x = 3 ), the term ( (x - 3)^2 cdot Q(x) ) becomes zero, so ( f(3) = R(3) ). Similarly, the first derivative of ( f(x) ) evaluated at ( x = 3 ) should equal the first derivative of ( R(x) ) evaluated at ( x = 3 ), because the derivative of ( (x - 3)^2 cdot Q(x) ) at ( x = 3 ) will also be zero.Let me compute ( f(3) ) first. Plugging in ( x = 3 ) into ( x^4 + 4 ):( f(3) = 3^4 + 4 = 81 + 4 = 85 ).So, ( R(3) = a cdot 3 + b = 3a + b = 85 ). That's one equation.Now, let's find the first derivative of ( f(x) ). The derivative of ( x^4 + 4 ) is ( 4x^3 ). So, ( f'(x) = 4x^3 ). Evaluating this at ( x = 3 ):( f'(3) = 4 cdot 3^3 = 4 cdot 27 = 108 ).On the other hand, the derivative of ( R(x) = ax + b ) is just ( a ). So, ( R'(3) = a ). Therefore, ( a = 108 ).Wait, that seems too big. Let me double-check. If ( R(x) = ax + b ), then ( R'(x) = a ), so ( R'(3) = a ). And since ( f'(3) = 108 ), that means ( a = 108 ). Hmm, okay, maybe that's correct.But let me think again. If ( f(x) = (x - 3)^2 Q(x) + R(x) ), then taking the derivative:( f'(x) = 2(x - 3) Q(x) + (x - 3)^2 Q'(x) + R'(x) ).At ( x = 3 ), the first two terms vanish because they have factors of ( (x - 3) ), so ( f'(3) = R'(3) = a ). So, yes, ( a = 108 ).Now, going back to the first equation, ( 3a + b = 85 ). Plugging in ( a = 108 ):( 3 cdot 108 + b = 85 )( 324 + b = 85 )Subtracting 324 from both sides:( b = 85 - 324 = -239 ).Wait, that gives ( R(x) = 108x - 239 ). But earlier, I thought the remainder was ( 31x - 56 ). Did I make a mistake somewhere?Let me try another approach to verify. Maybe I can perform polynomial long division. Let's see.Dividing ( x^4 + 4 ) by ( (x - 3)^2 ). First, note that ( (x - 3)^2 = x^2 - 6x + 9 ).So, we can set up the division:Divide ( x^4 + 0x^3 + 0x^2 + 0x + 4 ) by ( x^2 - 6x + 9 ).First term: ( x^4 ) divided by ( x^2 ) is ( x^2 ). Multiply ( x^2 ) by ( x^2 - 6x + 9 ):( x^2 cdot x^2 = x^4 )( x^2 cdot (-6x) = -6x^3 )( x^2 cdot 9 = 9x^2 )Subtracting this from the original polynomial:( (x^4 + 0x^3 + 0x^2 + 0x + 4) - (x^4 - 6x^3 + 9x^2) = 0x^4 + 6x^3 - 9x^2 + 0x + 4 ).Next term: ( 6x^3 ) divided by ( x^2 ) is ( 6x ). Multiply ( 6x ) by ( x^2 - 6x + 9 ):( 6x cdot x^2 = 6x^3 )( 6x cdot (-6x) = -36x^2 )( 6x cdot 9 = 54x )Subtracting this from the current polynomial:( (6x^3 - 9x^2 + 0x + 4) - (6x^3 - 36x^2 + 54x) = 0x^3 + 27x^2 - 54x + 4 ).Next term: ( 27x^2 ) divided by ( x^2 ) is ( 27 ). Multiply ( 27 ) by ( x^2 - 6x + 9 ):( 27 cdot x^2 = 27x^2 )( 27 cdot (-6x) = -162x )( 27 cdot 9 = 243 )Subtracting this from the current polynomial:( (27x^2 - 54x + 4) - (27x^2 - 162x + 243) = 0x^2 + 108x - 239 ).So, the remainder is ( 108x - 239 ). Wait, that's the same as before. But earlier, I thought the remainder was ( 31x - 56 ). Hmm, maybe I confused the problem with another one.Wait, let me check the initial problem again. It says to divide ( x^4 + 4 ) by ( (x - 3)^2 ). So, according to my calculations, the remainder is ( 108x - 239 ). But in the previous example, the remainder was ( 31x - 56 ). Maybe that was a different problem.Alternatively, perhaps I made a mistake in my derivative approach. Let me go back to that.We have ( f(x) = x^4 + 4 ), and we want to express it as ( (x - 3)^2 Q(x) + R(x) ), where ( R(x) = ax + b ).Then, ( f(3) = R(3) ) and ( f'(3) = R'(3) ).Calculating ( f(3) = 81 + 4 = 85 ), so ( 3a + b = 85 ).Calculating ( f'(x) = 4x^3 ), so ( f'(3) = 4*27 = 108 ). Since ( R'(x) = a ), then ( a = 108 ).Substituting back, ( 3*108 + b = 85 ) gives ( 324 + b = 85 ), so ( b = 85 - 324 = -239 ).So, ( R(x) = 108x - 239 ).But in the initial problem, the user had a different remainder, ( 31x - 56 ). Maybe they made a mistake, or perhaps I'm misunderstanding the problem.Wait, let me try another method. Maybe using the Taylor series expansion around x = 3. Since we're dividing by ( (x - 3)^2 ), the remainder can be found by expanding ( f(x) ) around x = 3 up to the linear term.So, ( f(x) = f(3) + f'(3)(x - 3) + frac{f''(3)}{2}(x - 3)^2 + dots ). But since we're only interested in the remainder when divided by ( (x - 3)^2 ), the terms beyond the linear term will be part of the quotient multiplied by ( (x - 3)^2 ). So, the remainder is ( f(3) + f'(3)(x - 3) ).Wait, that would be ( 85 + 108(x - 3) ). Let's compute that:( 85 + 108x - 324 = 108x - 239 ).So, that's consistent with my previous results. Therefore, the remainder should be ( 108x - 239 ).But in the initial problem, the user had a different answer, ( 31x - 56 ). Maybe they were solving a different problem or made a mistake in their calculation.Alternatively, perhaps I'm misunderstanding the problem. Let me check the initial problem again: "Find the remainder when ( x^4 + 4 ) is divided by ( (x - 3)^2 )."Yes, that's correct. So, according to my calculations, the remainder is ( 108x - 239 ). But the initial example had a different remainder. Maybe the initial example was incorrect.Alternatively, perhaps the user used a different method, like dividing twice by ( x - 3 ), which might lead to the same result but expressed differently.Wait, let me try that approach. If I first divide ( x^4 + 4 ) by ( x - 3 ), get a quotient and remainder, then divide that quotient by ( x - 3 ) again, and then combine the remainders.So, first division: ( x^4 + 4 ) divided by ( x - 3 ).Using synthetic division:3 | 1 0 0 0 4 3 9 27 81 1 3 9 27 85So, the quotient is ( x^3 + 3x^2 + 9x + 27 ) and the remainder is 85.Now, divide this quotient ( x^3 + 3x^2 + 9x + 27 ) by ( x - 3 ) again.Using synthetic division:3 | 1 3 9 27 3 18 81 1 6 27 108So, the quotient is ( x^2 + 6x + 27 ) and the remainder is 108.Therefore, combining these, we have:( x^4 + 4 = (x - 3)(x^3 + 3x^2 + 9x + 27) + 85 )And ( x^3 + 3x^2 + 9x + 27 = (x - 3)(x^2 + 6x + 27) + 108 )Substituting back:( x^4 + 4 = (x - 3)[(x - 3)(x^2 + 6x + 27) + 108] + 85 )Expanding:( x^4 + 4 = (x - 3)^2(x^2 + 6x + 27) + 108(x - 3) + 85 )Now, simplifying the constants:( 108(x - 3) + 85 = 108x - 324 + 85 = 108x - 239 )So, the remainder is ( 108x - 239 ), which matches my earlier result.Therefore, the correct remainder is ( 108x - 239 ). It seems that the initial example had a different remainder, possibly due to a miscalculation or a different problem.Wait, let me check the initial example again. The user wrote:"Find the remainder when ( x^4 + 4 ) is divided by ( (x - 3)^2 )."Then, their solution was:Firstly, divide ( x^4 + 4 ) by ( x - 3 ),[x^4 + 4 = (x - 3)(x^3 + 3x^2 + 9x + 27) + 31.]Then, divide ( x^3 + 3x^2 + 9x + 27 ) by ( x - 3 ),[x^3 + 3x^2 + 9x + 27 = (x - 3)(x^2 + 6x + 18) + 37.]Compiling these results:[x^4 + 4 = (x - 3)^2(x^2 + 6x + 18) + 31(x - 3) + 37.]Substituting ( (x - 3) ) back in and simplifying:[= (x - 3)^2(x^2 + 6x + 18) + 31x - 93 + 37.][= (x - 3)^2(x^2 + 6x + 18) + 31x - 56,]so the remainder is ( boxed{31x - 56} ).Wait, in their first division, they got a remainder of 31, but when I did the division, I got a remainder of 85. That's a discrepancy. Let me check their synthetic division.If they divided ( x^4 + 4 ) by ( x - 3 ), the remainder should be ( f(3) = 81 + 4 = 85 ). But they wrote the remainder as 31. That's incorrect. So, their initial step was wrong, leading to an incorrect remainder.Therefore, their final answer of ( 31x - 56 ) is incorrect because their first division was wrong. The correct remainder should be ( 108x - 239 ).So, in conclusion, the remainder when ( x^4 + 4 ) is divided by ( (x - 3)^2 ) is ( 108x - 239 ).

💡Okay, so I have this problem where I'm given that the cosine of theta is 3/5, and I need to find the cosine of 2 theta. Hmm, I remember there are some trigonometric identities that relate double angles to single angles. Let me try to recall them.First, I think the double angle formula for cosine is one of those identities. I believe it goes something like cos(2θ) equals 2 times cos squared theta minus 1. Yeah, that sounds right. So, if I can find cos squared theta, I can plug it into this formula and get the answer.Given that cos(theta) is 3/5, I can square that to find cos squared theta. So, (3/5) squared is 9/25. That seems straightforward. Now, plugging that back into the double angle formula: 2 times 9/25 minus 1. Let me calculate that step by step.First, 2 times 9/25 is 18/25. Then, subtracting 1 from that. But wait, 1 is the same as 25/25, right? So, 18/25 minus 25/25 is... hmm, that would be negative 7/25. So, cos(2θ) is negative 7/25.Wait a second, is that correct? Let me double-check my steps. I used the double angle formula correctly, right? Cos(2θ) equals 2 cos squared theta minus 1. Yes, that's the formula. And cos(theta) is 3/5, so cos squared theta is indeed 9/25. Multiplying that by 2 gives 18/25, and subtracting 1, which is 25/25, results in negative 7/25. Yeah, that seems right.But just to be thorough, maybe I should consider another method to verify. I remember there's another double angle formula that uses sine, like cos(2θ) equals 1 minus 2 sin squared theta. Maybe I can use that as a check.First, I need to find sin(theta). Since I know cos(theta) is 3/5, I can use the Pythagorean identity, which says that sin squared theta plus cos squared theta equals 1. So, sin squared theta is 1 minus cos squared theta, which is 1 minus 9/25. That gives 16/25. Therefore, sin(theta) is either 4/5 or -4/5, but since the problem doesn't specify the quadrant, I'll assume it's positive for simplicity.Now, using the other double angle formula: cos(2θ) equals 1 minus 2 sin squared theta. Plugging in sin squared theta as 16/25, we get 1 minus 2 times 16/25. That's 1 minus 32/25. Converting 1 to 25/25, we have 25/25 minus 32/25, which is negative 7/25. So, that's the same result as before. That makes me more confident that the answer is correct.Alternatively, I could also use the cosine of sum formula, where cos(2θ) is cos(theta + theta), which is cos(theta)cos(theta) minus sin(theta)sin(theta). That simplifies to cos squared theta minus sin squared theta. Plugging in the values, that's 9/25 minus 16/25, which is again negative 7/25. So, all three methods lead me to the same answer.I think I'm pretty sure now that cos(2θ) is negative 7/25. It's interesting how different identities can lead to the same result. I guess that's the beauty of trigonometry—there are multiple ways to approach a problem, and they all converge to the correct answer if done correctly.Wait, just thinking about the angle itself, if cos(theta) is 3/5, that means theta is in the first quadrant since cosine is positive. So, 2 theta would be in the second quadrant because doubling an angle in the first quadrant can take it into the second quadrant where cosine is negative. That makes sense because our answer is negative, so cos(2θ) is negative, which aligns with it being in the second quadrant. That's another way to confirm that the negative result is plausible.So, putting it all together, using the double angle formulas and verifying with multiple methods, I'm confident that cos(2θ) is negative 7/25.

💡First, identify the longest side of the triangle, which is 8 cm.Next, apply the Pythagorean theorem to determine the type of triangle:- Calculate the sum of the squares of the two shorter sides: (4^2 + 6^2 = 16 + 36 = 52).- Compare this sum to the square of the longest side: (8^2 = 64).Since (52 < 64), the sum of the squares of the two shorter sides is less than the square of the longest side. This indicates that the triangle is obtuse.Therefore, the triangle with sides 4 cm, 6 cm, and 8 cm is an obtuse triangle.