x^4-5x^3-16x^2+23x-3 divided by x^2+2x-3
x^2+2x-3 factors into (x+3)(x-1). You can divide x^4-5x^3-16x^2+23x-3 by (x+3) and (x-1) using synthetic division, which may be easier.
To divide the polynomial (x^4 - 5x^3 - 16x^2 + 23x - 3) by the polynomial (x^2 + 2x - 3), we can use polynomial long division. Here's how to do it step by step:
Step 1: Arrange the terms of both polynomials in descending order of exponents.
Dividend: x^4 - 5x^3 - 16x^2 + 23x - 3
Divisor: x^2 + 2x - 3
Step 2: Divide the first term of the dividend by the first term of the divisor.
The first term of the dividend is x^4, and the first term of the divisor is x^2. Dividing x^4 by x^2 gives x^2.
Step 3: Multiply the whole divisor by the quotient obtained in the previous step.
The quotient obtained in step 2 is x^2. Multiply the divisor (x^2 + 2x - 3) by x^2 to get x^4 + 2x^3 - 3x^2.
Step 4: Subtract the result obtained in step 3 from the dividend.
Subtracting (x^4 + 2x^3 - 3x^2) from the dividend (x^4 - 5x^3 - 16x^2 + 23x - 3) eliminates the first term of the dividend.
The subtraction result is: -7x^3 - 13x^2 + 23x - 3.
Step 5: Repeat steps 2-4 with the new polynomial obtained in the previous step.
New divisor: x^2 + 2x - 3
New dividend: -7x^3 - 13x^2 + 23x - 3
After dividing, you will get the final quotient and remainder:
Quotient: x^2 - 7x - 2
Remainder: 12x + 1
Therefore, (x^4 - 5x^3 - 16x^2 + 23x - 3) divided by (x^2 + 2x - 3) is equal to (x^2 - 7x - 2) with a remainder of (12x + 1).