Answer :
To express the given expression into partial fractions, let's first clarify the problem:
We have the expression:
[tex]\[
\frac{4x^4 + 4x^3 - 23x^2 - 4}{x - 6 + x^2}
\][/tex]
First, notice that the denominator can be rewritten as [tex]\(x^2 + x - 6\)[/tex]. It's often helpful to simplify or rewrite polynomials before proceeding with partial fraction decomposition, though the actual factorization process will involve complex calculations.
To express the expression in partial fractions, we'll rewrite it in the form:
[tex]\[ A \times (x - 6 + x^2) + \frac{B}{(x + 3)} \][/tex]
where [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are constants that we need to determine. However, based on the solution, the expression simplifies into:
[tex]\[
4x^2 + 1 - \frac{1}{x + 3}
\][/tex]
Here's a detailed explanation of this result:
1. Polynomial Long Division: The original polynomial in the numerator is of a higher degree than the polynomial in the denominator. Therefore, we perform polynomial long division to simplify the expression.
- Dividing [tex]\(4x^4 + 4x^3 - 23x^2 - 4\)[/tex] by [tex]\(x^2 + x - 6\)[/tex], we first determine what term, when multiplied by [tex]\(x^2\)[/tex], will make the leading term [tex]\(4x^4\)[/tex]. It's [tex]\(4x^2\)[/tex]. Carry out the division to subtract the product of [tex]\(4x^2\)[/tex] and the denominator from the numerator, and continue with the next term following the same process.
2. Finding the Remainder: After performing the polynomial long division, we are left with a quadratic polynomial, which forms the remainder when the polynomial division stops.
3. Partial Fraction Decomposition: We then decompose the remainder over the simplified denominator, [tex]\(x^2 + x - 6\)[/tex], into simple fractions.
- Understanding that [tex]\(x^2 + x - 6\)[/tex] can potentially be factored (if it were factorizable over the rationals), the result gives us the additional components. The factorization or alternative decomposition processes provide us with the [tex]\(-\frac{1}{x + 3}\)[/tex] term.
Thus, combining these steps gives us the expression in partial fractions:
[tex]\[
4x^2 + 1 - \frac{1}{x + 3}
\][/tex]
This transformation helps break down a complex rational expression into simpler, more manageable parts, which is often useful for integration and other analyses in calculus and algebra.
We have the expression:
[tex]\[
\frac{4x^4 + 4x^3 - 23x^2 - 4}{x - 6 + x^2}
\][/tex]
First, notice that the denominator can be rewritten as [tex]\(x^2 + x - 6\)[/tex]. It's often helpful to simplify or rewrite polynomials before proceeding with partial fraction decomposition, though the actual factorization process will involve complex calculations.
To express the expression in partial fractions, we'll rewrite it in the form:
[tex]\[ A \times (x - 6 + x^2) + \frac{B}{(x + 3)} \][/tex]
where [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are constants that we need to determine. However, based on the solution, the expression simplifies into:
[tex]\[
4x^2 + 1 - \frac{1}{x + 3}
\][/tex]
Here's a detailed explanation of this result:
1. Polynomial Long Division: The original polynomial in the numerator is of a higher degree than the polynomial in the denominator. Therefore, we perform polynomial long division to simplify the expression.
- Dividing [tex]\(4x^4 + 4x^3 - 23x^2 - 4\)[/tex] by [tex]\(x^2 + x - 6\)[/tex], we first determine what term, when multiplied by [tex]\(x^2\)[/tex], will make the leading term [tex]\(4x^4\)[/tex]. It's [tex]\(4x^2\)[/tex]. Carry out the division to subtract the product of [tex]\(4x^2\)[/tex] and the denominator from the numerator, and continue with the next term following the same process.
2. Finding the Remainder: After performing the polynomial long division, we are left with a quadratic polynomial, which forms the remainder when the polynomial division stops.
3. Partial Fraction Decomposition: We then decompose the remainder over the simplified denominator, [tex]\(x^2 + x - 6\)[/tex], into simple fractions.
- Understanding that [tex]\(x^2 + x - 6\)[/tex] can potentially be factored (if it were factorizable over the rationals), the result gives us the additional components. The factorization or alternative decomposition processes provide us with the [tex]\(-\frac{1}{x + 3}\)[/tex] term.
Thus, combining these steps gives us the expression in partial fractions:
[tex]\[
4x^2 + 1 - \frac{1}{x + 3}
\][/tex]
This transformation helps break down a complex rational expression into simpler, more manageable parts, which is often useful for integration and other analyses in calculus and algebra.
