answer:Here are some potential names for your platform, along with some tagline suggestions: 1. **Artmos** - Immerse yourself in the art of tomorrow Artmos suggests a fusion of art and atmosphere, implying a fully immersive experience. 2. **Kaleido** - Where art meets infinity Kaleido conveys a sense of vibrant colors, constant change, and endless possibilities, reflecting the diverse and dynamic nature of digital art. 3. **Exhibita** - Experience art without boundaries Exhibita emphasizes the idea of art exhibitions without physical limitations, inviting users to explore and interact with digital art in new ways. 4. **Artphere** - Enter the sphere of creativity Artphere suggests a self-contained, immersive environment where artists and audiences can connect, create, and inspire each other. 5. **Spectra** - Where art meets technology Spectra implies a convergence of artistic expression and technological innovation, highlighting the unique possibilities of digital art. 6. **Mirovia** - Reflecting the infinite possibilities of art Mirovia, derived from the word mirror, suggests a platform that reflects and amplifies the creative potential of artists and audiences alike. 7. **Canvasa** - Unleash your creativity on the digital canvas Canvasa emphasizes the idea of a virtual canvas, where artists can experiment, express themselves, and push the boundaries of digital art. 8. **Lumin** - Illuminating the art of the future Lumin conveys a sense of light, energy, and innovation, implying a platform that shines a light on the most exciting and innovative digital art. 9. **Artifex** - Where art meets imagination Artifex, derived from the Latin word for artist or craftsman, highlights the creative and imaginative potential of digital art. 10. **Nexart** - Connecting artists, art, and audience Nexart emphasizes the idea of connection and community, bringing together artists, art enthusiasts, and technology to create a vibrant and dynamic ecosystem. Feel free to mix and match names and taglines to find the perfect combination that resonates with your vision and mission.

answer:To compute the remainder when dividing f(x) by g(x), we'll use polynomial long division. We start by dividing the highest-degree term of f(x), which is 3x^4, by the highest-degree term of g(x), which is x^2. This gives us 3x^2, which we then multiply by g(x) to get 3x^4 - 3x^3 - 6x^2. We subtract this from f(x) to get a new polynomial: 3x^4 - 6x^3 + x + 5 - (3x^4 - 3x^3 - 6x^2) = -3x^3 + 6x^2 + x + 5. Now we repeat the process, dividing the highest-degree term of this new polynomial, which is -3x^3, by the highest-degree term of g(x), x^2. This gives us -3x, which we multiply by g(x) to get -3x^3 + 3x^2 + 6x. Subtracting this from our previous polynomial gives us: -3x^3 + 6x^2 + x + 5 - (-3x^3 + 3x^2 + 6x) = 3x^2 - 5x + 5. We continue this process by dividing 3x^2 by x^2 to get 3, then multiplying g(x) by 3 to get 3x^2 - 3x - 6. Subtracting this from our previous polynomial gives us: 3x^2 - 5x + 5 - (3x^2 - 3x - 6) = -2x + 11. Since -2x + 11 is of lower degree than g(x), we can stop the division process, and this is our remainder. Therefore, the remainder when dividing f(x) by g(x) is -2x + 11.

answer:We can simplify the process by focusing on the key steps and omitting the intermediate details. Divide f(x) = 3x^4 - 6x^3 + x + 5 by g(x) = x^2 - x - 2. Starting with the highest-degree term, 3x^4, divide by x^2 to get 3x^2, then multiply g(x) by 3x^2 and subtract from f(x). This leaves us with a polynomial that still has a higher degree than g(x), so we repeat the process until we obtain a polynomial of lower degree. Continuing in this manner, we eventually find that f(x) can be expressed as a multiple of g(x) plus a remainder. Working through the division, we ultimately obtain the remainder as -2x + 11. This result can be verified by realizing that g(x) is a quadratic function, so the remainder must be a linear function of the form ax + b.

answer:The remainder theorem states that if a polynomial f(x) is divided by (x - a), then the remainder is f(a). Although we're dividing f(x) by a quadratic g(x) = x^2 - x - 2, rather than a linear term (x - a), the remainder theorem provides insight into the division process. Notice that g(x) can be factored as (x - 2)(x + 1). According to the remainder theorem, if we divide f(x) by (x - 2), the remainder is f(2), and if we divide f(x) by (x + 1), the remainder is f(-1). While this doesn't directly give us the remainder when dividing by the quadratic g(x), it suggests that the remainder -2x + 11 is related to the behavior of f(x) at the roots of g(x), which are x = 2 and x = -1. In fact, we can verify the remainder by checking that f(2) = (-2)(2) + 11 = 7 and f(-1) = (-2)(-1) + 11 = 13 are consistent with the divisions f(x) / (x - 2) and f(x) / (x + 1), respectively. This connection highlights the relationship between the remainder theorem and polynomial division.