answer:To find the roots of p'(x), we first need to find the derivative of p(x). Using the power rule for differentiation, we obtain p'(x) = 4x^3 - 6x^2 + 6x - 4. Now, we need to find the values of x for which p'(x) equals zero. We notice that p'(x) can be factored as p'(x) = 2(2x^3 - 3x^2 + 3x - 2). However, factoring this cubic polynomial further is not straightforward. Instead, we can try to find rational roots using the Rational Root Theorem. This theorem states that if a rational number p/q is a root of the polynomial, then p must be a factor of the constant term and q must be a factor of the leading coefficient. In this case, the constant term is -2, and the leading coefficient is 2. The possible rational roots are therefore ±1 and ±2. We can test these values by substituting them into p'(x). Upon testing, we find that x = 1 is a root of p'(x). We can now perform polynomial division to factor p'(x) as p'(x) = 2(x-1)(2x^2 - x + 2). The quadratic term 2x^2 - x + 2 does not factor further over the real numbers, but we can find its roots using the quadratic formula. The roots are (1 ± √(-15))/4, which are complex numbers. So, the roots of p'(x) are x = 1 and x = (1 ± i√15)/4.

answer:We start by simplifying the equation by dividing both sides by the greatest common divisor (GCD) of 30 and 12, which is 6. This yields 5x + 2y = 3. Now we can try to find integer solutions to this equation. Since 5 and 2 are relatively prime (their GCD is 1), we can use the Euclidean algorithm to find a particular solution. Notice that 5(1) + 2(-2) = 1, so multiplying both sides by 3 gives us 5(3) + 2(-6) = 3. Therefore, a particular solution is x = 3, y = -6. However, we are interested in finding all integer solutions. We can write the general solution as x = 3 + 2k, y = -6 - 5k, where k is an integer. This is because adding 2k to x and subtracting 5k from y preserves the equality, since 5(2k) - 2(5k) = 0. So, the integer solutions to the Diophantine equation 30x + 12y = 18 are given by x = 3 + 2k, y = -6 - 5k, for any integer k.

answer:We notice that 7^x = 8^x - 1 can be written as (8-1)^x = 8^x - 1. This suggests that the equation might be related to the binomial expansion of (8-1)^x. We can try to rewrite (8-1)^x using the binomial theorem. Since 7 = 8 - 1, we have: (8-1)^x = 8^x - x*8^(x-1) + ... - 1 where the ellipsis represents terms with lower powers of 8. Comparing this to the original equation, we see that the -1 term matches, but the middle terms seem to be missing. However, the key insight is that these middle terms are all non-negative, since they involve powers of 8 multiplied by binomial coefficients. Therefore, for the equation 7^x = 8^x - 1 to hold, the middle terms must sum to zero. This can only happen if x is 1, because for x > 1, the middle terms would be positive. For x = 1, we have 7^1 = 8^1 - 1, which is indeed true. We can also show that x = 1 is the only solution by using the fact that 7^x grows slower than 8^x as x increases. Specifically, for x > 1, we have 7^x < 8^x - 1, since the difference between 7^x and 8^x grows exponentially with x. So, the only solution to the equation 7^x = 8^x - 1 is x = 1.

answer:A platypus and a duck-billed platypus are actually the same creature. The term duck-billed platypus is often used to describe the platypus due to its distinctive bill, which resembles a duck's bill. This unique feature, along with its webbed feet and egg-laying abilities, make the platypus one of the most recognizable and fascinating creatures in the animal kingdom. The platypus is a mammal that belongs to the monotreme group, a distinct group of mammals that lay eggs instead of giving birth to live young like most other mammals. Its scientific name, Ornithorhynchus anatinus, reflects its unique characteristics, with Ornithorhynchus meaning bird-snout and anatinus meaning duck-like. So, in essence, the terms platypus and duck-billed platypus are often used interchangeably to describe this remarkable animal.