💡Alright, so I have this polynomial: (3x^3 - dx + 18), and it's supposed to have a factor of the form (x^2 + qx + 2). I need to find the value of (d). Hmm, okay, let me think about how to approach this.First, I remember that if a polynomial has a factor, then it can be expressed as the product of that factor and another polynomial. Since the given polynomial is a cubic (degree 3) and the factor is quadratic (degree 2), the other factor must be linear (degree 1). So, I can write:[3x^3 - dx + 18 = (x^2 + qx + 2)(mx + n)]Here, (mx + n) is the linear factor I need to find. My goal is to determine the values of (m), (n), (q), and (d). But since I only need to find (d), maybe I can express everything in terms of (d) and solve for it.Let me expand the right-hand side:[(x^2 + qx + 2)(mx + n) = mx^3 + (n + mq)x^2 + (2m + nq)x + 2n]Now, let's write down the original polynomial:[3x^3 - dx + 18]I notice that there's no (x^2) term in the original polynomial, which means the coefficient of (x^2) must be zero. Similarly, the coefficient of (x^3) is 3, and the constant term is 18. Let's set up equations by equating the coefficients from both sides.1. Coefficient of (x^3): (m = 3)2. Coefficient of (x^2): (n + mq = 0)3. Coefficient of (x): (2m + nq = -d)4. Constant term: (2n = 18)Alright, let's solve these step by step.From equation 1: (m = 3). That was straightforward.From equation 4: (2n = 18), so (n = 9). Easy enough.Now, plug (m = 3) and (n = 9) into equation 2: (9 + 3q = 0). Solving for (q):[3q = -9 q = -3]Okay, so (q = -3). Now, let's move to equation 3: (2m + nq = -d). Plugging in the known values:[2(3) + 9(-3) = -d 6 - 27 = -d -21 = -d d = 21]Wait, hold on. Let me double-check that calculation. So, (2m) is 6, and (nq) is (9 times (-3) = -27). Adding them together: (6 + (-27) = -21). So, (-21 = -d), which means (d = 21). Hmm, that seems correct.But let me verify by plugging everything back into the original equation to make sure I didn't make a mistake.So, if (m = 3), (n = 9), and (q = -3), then the quadratic factor is (x^2 - 3x + 2), and the linear factor is (3x + 9). Let's multiply them:[(x^2 - 3x + 2)(3x + 9)]Expanding this:First, multiply (x^2) by each term in (3x + 9):- (x^2 times 3x = 3x^3)- (x^2 times 9 = 9x^2)Next, multiply (-3x) by each term in (3x + 9):- (-3x times 3x = -9x^2)- (-3x times 9 = -27x)Then, multiply (2) by each term in (3x + 9):- (2 times 3x = 6x)- (2 times 9 = 18)Now, add all these together:[3x^3 + 9x^2 - 9x^2 - 27x + 6x + 18]Combine like terms:- (3x^3)- (9x^2 - 9x^2 = 0)- (-27x + 6x = -21x)- (18)So, the result is:[3x^3 - 21x + 18]Which matches the original polynomial (3x^3 - dx + 18) if (d = 21). Therefore, my calculation seems correct.Wait, but in the initial problem statement, the polynomial is written as (3x^3 - dx + 18). So, the coefficient of (x) is (-d), and in our expanded polynomial, it's (-21x). Therefore, (-d = -21), so (d = 21). Yep, that checks out.I think I've got it right. So, the value of (d) is 21.**Final Answer**The value of (d) is boxed{21}.

💡Okay, so I need to find the least common multiple (LCM) of the numbers 28 and 72. Hmm, I remember that the LCM is the smallest number that both 28 and 72 can divide into without leaving a remainder. But I'm not entirely sure how to calculate it step by step. Let me think about the methods I know for finding the LCM.One way I recall is using prime factorization. Yeah, that sounds right. So, first, I need to break down both numbers into their prime factors. Let me try that.Starting with 28. I know that 28 is an even number, so it's divisible by 2. Dividing 28 by 2 gives me 14. Okay, 14 is also even, so I can divide by 2 again to get 7. Now, 7 is a prime number, so I can't break it down further. So, the prime factors of 28 are 2, 2, and 7. In exponential form, that would be 2 squared times 7, or 2² × 7.Now, moving on to 72. This is a bit bigger, but I think I can handle it. 72 is also even, so dividing by 2 gives me 36. 36 is still even, so dividing by 2 again gives me 18. 18 is also even, so dividing by 2 once more gives me 9. Now, 9 is not even, but it's divisible by 3. Dividing 9 by 3 gives me 3, and then dividing by 3 again gives me 1. So, the prime factors of 72 are 2, 2, 2, 3, and 3. In exponential form, that's 2 cubed times 3 squared, or 2³ × 3².Alright, so now I have the prime factors:- 28 = 2² × 7- 72 = 2³ × 3²To find the LCM, I think I need to take the highest power of each prime number that appears in the factorizations. So, for the prime number 2, the highest power is 2³ from 72. For the prime number 3, the highest power is 3² from 72. And for the prime number 7, the highest power is 7¹ from 28.So, putting that all together, the LCM should be 2³ × 3² × 7. Let me calculate that step by step. First, 2³ is 8. Then, 3² is 9. Multiplying those together, 8 × 9 equals 72. Finally, multiplying that result by 7 gives me 72 × 7, which is 504.Wait, does that make sense? Let me double-check. If I list the multiples of 28 and 72, do they both divide 504?Multiples of 28: 28, 56, 84, 112, 140, 168, 196, 224, 252, 280, 308, 336, 364, 392, 420, 448, 476, 504, ...Multiples of 72: 72, 144, 216, 288, 360, 432, 504, ...Yes, 504 is the first common multiple they both share. So, that seems correct.I guess another way to find the LCM is by using the formula: LCM(a, b) = |a × b| / GCD(a, b). Do I remember how to find the greatest common divisor (GCD)? Let me try that method too to confirm.First, I need to find the GCD of 28 and 72. One way to do that is by listing the factors of each number and finding the largest one they have in common.Factors of 28: 1, 2, 4, 7, 14, 28.Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.The common factors are 1, 2, and 4. The greatest one is 4. So, GCD(28, 72) is 4.Now, using the formula: LCM(28, 72) = (28 × 72) / 4.Calculating that: 28 × 72 is... let's see, 28 × 70 is 1960, and 28 × 2 is 56, so adding them together gives 1960 + 56 = 2016. Then, dividing by 4: 2016 / 4 = 504.Okay, that matches the result I got earlier using prime factorization. So, that's reassuring. Both methods give me 504 as the LCM of 28 and 72.I think I've got it then. The least common multiple of 28 and 72 is 504.

💡To compute (1007^2 - 993^2 - 1005^2 + 995^2) without a calculator, I can use the difference of squares formula, which states that (a^2 - b^2 = (a + b)(a - b)).First, I'll apply this formula to the first pair of terms, (1007^2 - 993^2):[1007^2 - 993^2 = (1007 + 993)(1007 - 993)]Calculating the sum and difference:[1007 + 993 = 2000][1007 - 993 = 14]So, the first part becomes:[2000 times 14 = 28000]Next, I'll apply the difference of squares formula to the second pair of terms, (1005^2 - 995^2):[1005^2 - 995^2 = (1005 + 995)(1005 - 995)]Calculating the sum and difference:[1005 + 995 = 2000][1005 - 995 = 10]So, the second part becomes:[2000 times 10 = 20000]Finally, I'll subtract the second result from the first:[28000 - 20000 = 8000]

💡Okay, so I have this problem here: Lily has 4 gallons of milk and gives 16/3 gallons to James. I need to find out what fraction of a gallon she has left. Hmm, let's see. First, I know that Lily starts with 4 gallons. That's a whole number, but she gives away a fraction, which is 16/3 gallons. I think I need to subtract the amount she gives away from the total she has to find out how much is left. So, the basic idea is to do 4 minus 16/3.But wait, 4 is a whole number, and 16/3 is a fraction. To subtract them, it might be easier if both are fractions with the same denominator. I remember that to subtract fractions, they need to have the same denominator. So, I should convert 4 into a fraction with the same denominator as 16/3, which is 3.Alright, how do I convert 4 into a fraction with denominator 3? Well, 4 is the same as 4/1. To get a denominator of 3, I need to multiply both the numerator and the denominator by 3. So, 4/1 times 3/3 is 12/3. Okay, so 4 gallons is the same as 12/3 gallons.Now, I can subtract 16/3 from 12/3. Let's do that: 12/3 minus 16/3. When you subtract fractions with the same denominator, you just subtract the numerators and keep the denominator the same. So, 12 minus 16 is -4, and the denominator remains 3. So, that gives me -4/3.Wait a minute, that's a negative number. Does that make sense? If Lily gives away more milk than she has, she would end up owing milk, right? So, a negative result indicates that she doesn't have any milk left but actually needs to get more milk to fulfill the amount she gave to James.But the question asks for the fraction of a gallon she has left. If she has a negative amount, does that mean she has no milk left and actually owes some? Maybe I should interpret this as she has 0 gallons left and owes 4/3 gallons. But the problem specifically asks for the fraction of a gallon she has left, not the amount she owes.Hmm, perhaps I made a mistake in the subtraction. Let me double-check. She has 4 gallons, which is 12/3, and she gives away 16/3. So, 12/3 minus 16/3 is indeed -4/3. That seems correct mathematically. But in real life, you can't have negative milk, so maybe the problem expects me to recognize that she can't give away more than she has.Wait, maybe I should interpret this differently. Perhaps the problem assumes that she can give away more than she has, and we're just calculating the mathematical difference, regardless of the real-world implications. In that case, the answer would be -4/3 gallons, which indicates she has a deficit of 4/3 gallons.But the question asks for the fraction of a gallon she has left. So, if she has a negative amount, does that mean she has no milk left? Or should I present the answer as a positive fraction, indicating the amount she owes?I think the problem is expecting a positive fraction, so maybe I should take the absolute value of the result. That would make it 4/3 gallons. So, Lily has 4/3 gallons left, but since it's negative, it means she actually owes that amount. But the question is about what she has left, not what she owes.This is a bit confusing. Maybe I should consider that she can't give away more than she has, so she gives away all 4 gallons, and she has 0 left. But the problem says she gives 16/3 gallons, which is approximately 5.33 gallons, which is more than 4. So, she can't give that much.Alternatively, perhaps the problem is designed to show that she ends up owing milk, so the answer is -4/3 gallons. But since the question asks for the fraction of a gallon she has left, and not the amount she owes, maybe the answer should be 0 gallons, but that doesn't make sense because she gave away more than she had.Wait, perhaps I should present the answer as a positive fraction, indicating the amount she needs to get to have the amount she gave away. So, she has -4/3 gallons, which means she needs 4/3 gallons more to have the amount she gave to James. But the question is about what she has left, not what she needs.I'm getting a bit stuck here. Let me try to approach it differently. Maybe I should convert both quantities to decimals to see what's happening. 4 gallons is 4.0, and 16/3 is approximately 5.333... So, 4.0 minus 5.333 is approximately -1.333, which is -4/3. So, that confirms my earlier calculation.But again, the question is about the fraction of a gallon she has left. If she has a negative amount, does that mean she has no milk left and actually owes some? Or is the answer supposed to be a positive fraction, indicating the amount she has left?I think the problem is expecting a positive fraction, so maybe I should interpret the negative result as her owing that amount, but the question is about what she has left. So, perhaps the answer is 0 gallons, but that doesn't seem right because she gave away more than she had.Wait, maybe the problem is designed to show that she can't give away more than she has, so she gives away all 4 gallons, and she has 0 left. But the problem specifically says she gives 16/3 gallons, which is more than 4. So, that doesn't make sense.Alternatively, maybe the problem is testing the understanding of fractions and negative results, so the answer is -4/3 gallons, indicating she has a deficit. But the question asks for the fraction of a gallon she has left, not the deficit.I'm not sure. Maybe I should go with the mathematical result, which is -4/3 gallons, but present it as a fraction she has left, even though it's negative. Or perhaps the problem expects me to recognize that she can't give away more than she has, so she has 0 left.Wait, let me think again. If she has 4 gallons and gives away 16/3 gallons, which is 5 and 1/3 gallons, she can't give away more than she has. So, she gives away all 4 gallons, and she has 0 left. But the problem says she gives 16/3 gallons, so maybe the answer is 0 gallons left.But that doesn't seem right because the problem is asking for a fraction, and 0 is a whole number. Maybe I should present it as 0/1, but that's not helpful.Alternatively, perhaps the problem is designed to have a negative result, indicating she owes milk. So, the answer is -4/3 gallons, but the question is about what she has left, so maybe it's better to say she has no milk left and owes 4/3 gallons.But the question specifically asks for the fraction of a gallon she has left, not what she owes. So, maybe the answer is 0 gallons, but that's not a fraction.Wait, maybe I should think of it differently. If she gives away 16/3 gallons from 4 gallons, which is 12/3, then 12/3 minus 16/3 is -4/3. So, she has -4/3 gallons left, which means she owes 4/3 gallons.But the question is about what she has left, not what she owes. So, maybe the answer is 0 gallons, but that's not a fraction. Alternatively, the answer is -4/3 gallons, indicating she has a deficit.I'm not sure. Maybe the problem expects the answer to be 4/3 gallons, but that would mean she has more milk left, which doesn't make sense because she gave away more than she had.Wait, perhaps I should present the answer as 4/3 gallons, but with a note that it's a deficit. But the question is just asking for the fraction of a gallon she has left, so maybe it's better to present it as -4/3 gallons.Alternatively, maybe the problem is designed to have a positive result, so I should take the absolute value, making it 4/3 gallons left. But that doesn't make sense because she gave away more than she had.I'm really confused now. Let me try to summarize:- Lily has 4 gallons, which is 12/3 gallons.- She gives away 16/3 gallons.- 12/3 minus 16/3 is -4/3 gallons.So, mathematically, she has -4/3 gallons left, which means she owes 4/3 gallons. But the question is about what she has left, not what she owes. So, maybe the answer is 0 gallons, but that's not a fraction.Alternatively, the answer is -4/3 gallons, indicating a deficit. But the question asks for the fraction of a gallon she has left, so maybe it's better to present it as -4/3 gallons.I think I'll go with that, even though it's negative, because that's the mathematical result. So, Lily has -4/3 gallons left, which means she owes 4/3 gallons.

💡First, I need to determine the number of Sundays in a 30-day month that starts on a Sunday. Since there are 7 days in a week, the month will have 4 full weeks, which accounts for 28 days, and 2 extra days. Therefore, there will be 4 Sundays.Next, I'll calculate the total number of visitors on Sundays and the remaining days. On Sundays, there are 140 visitors per day, so for 4 Sundays, the total visitors are 140 multiplied by 4, which equals 560 visitors.For the other 26 days, there are 80 visitors per day. Multiplying 80 by 26 gives 2,080 visitors.Adding the visitors from Sundays and the other days together, the total number of visitors in the month is 560 plus 2,080, which equals 2,640 visitors.Finally, to find the average number of visitors per day, I'll divide the total number of visitors by the number of days in the month. So, 2,640 divided by 30 equals 88 visitors per day.

💡Okay, so I have this problem: factor ( x^4 - 81 ) as far as possible, where the factors are monic polynomials with real coefficients. Hmm, let me think about how to approach this.First, I remember that ( x^4 - 81 ) looks like a difference of squares because both ( x^4 ) and 81 are perfect squares. Specifically, ( x^4 ) is ( (x^2)^2 ) and 81 is ( 9^2 ). So, I can use the difference of squares formula, which is ( a^2 - b^2 = (a - b)(a + b) ).Applying that here, I get:[ x^4 - 81 = (x^2)^2 - 9^2 = (x^2 - 9)(x^2 + 9) ]Alright, so now I have two factors: ( x^2 - 9 ) and ( x^2 + 9 ). Let me check if these can be factored further.Starting with ( x^2 - 9 ), that's another difference of squares because ( x^2 ) is ( x^2 ) and 9 is ( 3^2 ). So, applying the difference of squares formula again:[ x^2 - 9 = (x - 3)(x + 3) ]Great, so now I have ( (x - 3)(x + 3) ) as factors. What about ( x^2 + 9 )? Hmm, that's a sum of squares. I remember that over real numbers, sums of squares can't be factored further because they don't have real roots. If I try to factor it, I would end up with complex numbers, which aren't allowed here since we need real coefficients.So, putting it all together, the factorization of ( x^4 - 81 ) is:[ (x - 3)(x + 3)(x^2 + 9) ]Let me double-check to make sure I didn't miss anything. Starting from the original expression:1. Recognize it as a difference of squares.2. Factor it into ( (x^2 - 9)(x^2 + 9) ).3. Further factor ( x^2 - 9 ) into ( (x - 3)(x + 3) ).4. Confirm that ( x^2 + 9 ) can't be factored further with real coefficients.Everything seems to check out. I don't think there are any other real factors possible here. So, I think this is the most factored form with monic polynomials and real coefficients.**Final Answer**The factored form of ( x^4 - 81 ) is boxed{(x - 3)(x + 3)(x^2 + 9)}.