Certainly! Let's go through the detailed steps of solving the given division of a polynomial by a linear expression.
We are given the expression [tex]\((-3x^4 + 7x^2 + 1) \div (x - 1)\)[/tex].
Step 1: Write down the polynomial and the divisor.
- Polynomial: [tex]\(-3x^4 + 7x^2 + 1\)[/tex]
- Divisor: [tex]\(x - 1\)[/tex]
Step 2: Set up the long division for polynomials.
Firstly, we arrange both the polynomial and the divisor in the standard form (descending powers of [tex]\(x\)[/tex]):
[tex]\[
\frac{-3x^4 + 0x^3 + 7x^2 + 0x + 1}{x - 1}
\][/tex]
Step 3: Divide the first term of the numerator by the first term of the divisor to get the first term of the quotient.
[tex]\[
\frac{-3x^4}{x} = -3x^3
\][/tex]
Step 4: Multiply the divisor by this term and subtract from the polynomial.
[tex]\[
(-3x^3)(x - 1) = -3x^4 + 3x^3
\][/tex]
[tex]\[
\left(-3x^4 + 0x^3 + 7x^2 + 0x + 1\right) - \left(-3x^4 + 3x^3\right) = 0x^4 - 3x^3 + 7x^2 + 0x + 1
\][/tex]
Step 5: Repeat the process with the new polynomial.
Now we proceed with the polynomial [tex]\(-3x^3 + 7x^2 + 0x + 1\)[/tex].
[tex]\[
\frac{-3x^3}{x} = -3x^2
\][/tex]
[tex]\[
(-3x^2)(x - 1) = -3x^3 + 3x^2
\][/tex]
[tex]\[
\left(-3x^3 + 7x^2 + 0x + 1\right) - \left(-3x^3 + 3x^2\right) = 0x^3 + 4x^2 + 0x + 1
\][/tex]
Step 6: Continue the process with the remaining polynomial.
Next, we use the polynomial [tex]\(4x^2 + 0x + 1\)[/tex].
[tex]\[
\frac{4x^2}{x} = 4x
\][/tex]
[tex]\[
(4x)(x - 1) = 4x^2 - 4x
\][/tex]
[tex]\[
\left(4x^2 + 0x + 1\right) - \left(4x^2 - 4x\right) = 4x + 1
\][/tex]
Step 7: One last step with the remaining polynomial.
Finally, we consider the polynomial [tex]\(4x + 1\)[/tex].
[tex]\[
\frac{4x}{x} = 4
\][/tex]
[tex]\[
(4)(x - 1) = 4x - 4
\][/tex]
[tex]\[
\left(4x + 1\right) - \left(4x - 4\right) = 5
\][/tex]
Step 8: No further division is possible because the remainder degree [tex]\(5\)[/tex] is less than the divisor's degree [tex]\(1\)[/tex].
Therefore, the division of [tex]\((-3x^4 + 7x^2 + 1)\)[/tex] by [tex]\((x - 1)\)[/tex] yields a polynomial plus a remainder. However, considering the simplified form where no further reduction is made, the expression remains as:
[tex]\[
\boxed{\frac{-3x^4 + 7x^2 + 1}{x - 1}}
\][/tex]
