A(x) = 5x3 -4x2+8x -2:B(x) = x2-2x+3
To perform polynomial long division for the polynomials \( A(x) = 5x^3 - 4x^2 + 8x - 2 \) and \( B(x) = x^2 - 2x + 3 \), follow these steps:
Divide the leading term of \( A(x) \) by the leading term of \( B(x) \): \[ \frac{5x^3}{x^2} = 5x \] So, the first term of the quotient is \( 5x \).
Multiply \( 5x \) by \( B(x) \) and subtract from \( A(x) \): \[ 5x \cdot B(x) = 5x(x^2 - 2x + 3) = 5x^3 - 10x^2 + 15x \]
Now, subtract this from \( A(x) \): \[ (5x^3 - 4x^2 + 8x - 2) - (5x^3 - 10x^2 + 15x) = (-4x^2 + 10x^2) + (8x - 15x) - 2 \] This simplifies to: \[ 6x^2 - 7x - 2 \]
Now, repeat the division process: Divide the leading term \( 6x^2 \) by the leading term \( x^2 \): \[ \frac{6x^2}{x^2} = 6 \] So, the next term of the quotient is \( +6 \).
Multiply \( 6 \) by \( B(x) \) and subtract from the previous remainder: \[ 6 \cdot B(x) = 6(x^2 - 2x + 3) = 6x^2 - 12x + 18 \] Now, subtract this from \( 6x^2 - 7x - 2 \): \[ (6x^2 - 7x - 2) - (6x^2 - 12x + 18) = (-7x + 12x) + (-2 - 18) \] This simplifies to: \[ 5x - 20 \]
Now, we cannot divide further as the degree of the remainder \( (5x - 20) \) is less than the degree of \( B(x) \).
Conclusion
The result of the polynomial division is: \[ A(x) \div B(x) = 5x + 6 + \frac{5x - 20}{x^2 - 2x + 3} \]
Thus, you can represent it as: \[ A(x) = B(x) \cdot (5x + 6) + (5x - 20) \]
