Let's consider two polynomial expressions: \( P(x) = 2x^4 - 3x^3 + 5x \) and \( Q(x) = x^3 + 2x^2 - 4 \). The first polynomial, \( P(x) \), is a fourth-degree polynomial because the highest exponent of \( x \) is 4. The second polynomial, \( Q(x) \), has a degree of three, as its highest exponent is 3. These two polynomials both comprise at least three terms, fulfilling the requirement for variety in both complexity and degree.
Now, let's perform operations involving both polynomials. For the sum \( P(x) + Q(x) \), we combine like terms to get \( 2x^4 - 3x^3 + 5x + x^3 + 2x^2 - 4 = 2x^4 - 2x^3 + 2x^2 + 5x - 4 \). The difference \( P(x) - Q(x) \) simplifies to \( (2x^4 - 3x^3 + 5x) - (x^3 + 2x^2 - 4) = 2x^4 - 4x^3 - 2x^2 + 5x + 4 \). The product \( P(x) \times Q(x) \) is more involved: multiplying out the terms results in a polynomial of degree 7. Lastly, for the quotient \( \frac{P(x)}{Q(x)} \), we divide \( P(x) \) by \( Q(x) \), where the quotient will be a polynomial of degree 1 (as the degrees of polynomials are subtracted). The result of the polynomial division can be expressed in a simplified form, depending on the remainder.
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