To divide the polynomial [tex]\( 5x^3 + 7x^2 - 26x \)[/tex] by [tex]\( x^2 + 2x - 4 \)[/tex] using polynomial long division, we follow these steps:
1. Setup and Initial Division:
Start by writing the division of polynomials in a long division format, similar to how you would with numbers.
- Dividend (numerator): [tex]\( 5x^3 + 7x^2 - 26x \)[/tex]
- Divisor (denominator): [tex]\( x^2 + 2x - 4 \)[/tex]
The term we need to focus on initially is [tex]\( \frac{5x^3}{x^2} = 5x \)[/tex].
2. First Multiplier:
Multiply the entire divisor by this initial term [tex]\( 5x \)[/tex]:
[tex]\[
(x^2 + 2x - 4) \cdot 5x = 5x^3 + 10x^2 - 20x
\][/tex]
3. First Subtraction:
Subtract this result from the original dividend:
[tex]\[
(5x^3 + 7x^2 - 26x) - (5x^3 + 10x^2 - 20x)
\][/tex]
Simplify:
[tex]\[
5x^3 + 7x^2 - 26x - 5x^3 - 10x^2 + 20x = -3x^2 - 6x
\][/tex]
4. Second Division:
Repeat the process with the new polynomial [tex]\( -3x^2 - 6x \)[/tex].
- Divide the first term of the result by the first term of the divisor: [tex]\( \frac{-3x^2}{x^2} = -3 \)[/tex].
5. Second Multiplier:
Multiply the entire divisor by [tex]\(-3\)[/tex]:
[tex]\[
(x^2 + 2x - 4) \cdot (-3) = -3x^2 - 6x + 12
\][/tex]
6. Second Subtraction:
Subtract this result from [tex]\( -3x^2 - 6x \)[/tex]:
[tex]\[
(-3x^2 - 6x) - (-3x^2 - 6x + 12)
\][/tex]
Simplify:
[tex]\[
-3x^2 - 6x + 3x^2 + 6x - 12 = -12
\][/tex]
Each coefficient is calculated and aligned as per the typical polynomial long division steps, and the quotient and remainder are obtained after these steps.
7. Result:
The quotient from this division is [tex]\( 5x - 3 \)[/tex], and the remainder is [tex]\(-12\)[/tex].
Thus, the division of the polynomial [tex]\( 5x^3 + 7x^2 - 26x \)[/tex] by [tex]\( x^2 + 2x - 4 \)[/tex] yields:
[tex]\[
\boxed{5x - 3 \quad \text{with a remainder of} \quad -12}
\][/tex]
