Which polynomial has factors of 4x^7 and x^4?

To find a polynomial that has 4x⁷ and x⁴ as factors, we start by considering the multiplication of these factors. When we multiply the two given polynomials together, we need to remember that:

• Each factor contributes to the overall degree of the polynomial.
• The coefficients in front of each term will also multiply together.

So, let’s perform the multiplication:

4x⁷ can be considered as 4 * x⁷ and x⁴ can be considered as 1 * x⁴. When we multiply these two terms together, we get:

4x⁷ * x⁴ = 4 * x⁽⁷⁺⁴⁾ = 4x¹¹

This means that the polynomial we are looking for is:

4x¹¹

Thus, a polynomial that has both 4x⁷ and x⁴ as factors is simply 4x¹¹. This polynomial is of degree 11, and if we want to express it as a complete polynomial, it can also be written as:

4x¹¹ + 0x¹⁰ + 0x⁹ + 0x⁸ + 0x⁶ + 0x⁵ + 0x⁴ + 0x³ + 0x² + 0x + 0

In conclusion, the polynomial is 4x¹¹, which confirms that both given factors are indeed part of its makeup.