What is the quotient of the polynomial expression 2x^3 – 13x^2 + 16x – 5 divided by x – 5?

Finding the Quotient of a Polynomial Division

To divide the polynomial 2x³ – 13x² + 16x – 5 by x – 5, we can use polynomial long division.

Step-by-Step Solution:

1. Set Up the Division: Write the expression in long division format: Divide 2x³ – 13x² + 16x – 5 by x – 5.
2. First Division Step: Divide the leading term of the dividend, 2x³, by the leading term of the divisor, x. This gives us 2x².
3. Multiply: Multiply 2x² by the entire divisor (x – 5), which results in 2x³ – 10x².
4. Subtract: Subtract (2x³ – 10x²) from (2x³ – 13x² + 16x – 5). This simplifies to -3x² + 16x – 5.
5. Next Step: Now divide the leading term -3x² by the leading term of the divisor x. This gives us -3x.
6. Multiply: Multiply -3x by the divisor (x – 5) which results in -3x² + 15x.
7. Subtract Again: Subtract (-3x² + 15x) from (-3x² + 16x – 5). This gives x – 5.
8. Final Division: Now divide x by x, giving us 1.
9. Multiply: Multiply 1 by the divisor (x – 5), giving us x – 5.
10. Final Subtraction: Subtract (x – 5) from (x – 5), which gives the remainder 0.

The Result:

So, the quotient of 2x³ – 13x² + 16x – 5 divided by x – 5 is 2x² – 3x + 1.