Answer :
We start with the equation
$$
7x^6 - 28x^5 + 35x^4 = 48.
$$
Step 1. Write the equation in standard form by bringing all terms to one side:
$$
7x^6 - 28x^5 + 35x^4 - 48 = 0.
$$
Step 2. Look for any common factors among all terms. Notice that although the first three terms have a common factor, the constant term $-48$ does not share any factor with them other than $1$, so there is no non‐trivial common factor that can be factored out of the entire polynomial.
Step 3. Next, we try to factor the polynomial further. One might observe that the first three terms can be grouped by factoring out $7x^4$, which gives
$$
7x^4(x^2 - 4x + 5) - 48.
$$
However, the term $-48$ is separate and does not allow further grouping with the factored part. After attempting to work with the structure of the polynomial, we find that no factorization into lower degree polynomials with integer or simple rational coefficients is possible.
Thus, the polynomial in its completely factored form (over the integers and rationals) is
$$
7x^6 - 28x^5 + 35x^4 - 48.
$$
This is the final answer in factored form.
$$
7x^6 - 28x^5 + 35x^4 = 48.
$$
Step 1. Write the equation in standard form by bringing all terms to one side:
$$
7x^6 - 28x^5 + 35x^4 - 48 = 0.
$$
Step 2. Look for any common factors among all terms. Notice that although the first three terms have a common factor, the constant term $-48$ does not share any factor with them other than $1$, so there is no non‐trivial common factor that can be factored out of the entire polynomial.
Step 3. Next, we try to factor the polynomial further. One might observe that the first three terms can be grouped by factoring out $7x^4$, which gives
$$
7x^4(x^2 - 4x + 5) - 48.
$$
However, the term $-48$ is separate and does not allow further grouping with the factored part. After attempting to work with the structure of the polynomial, we find that no factorization into lower degree polynomials with integer or simple rational coefficients is possible.
Thus, the polynomial in its completely factored form (over the integers and rationals) is
$$
7x^6 - 28x^5 + 35x^4 - 48.
$$
This is the final answer in factored form.
