how is dividing a polynomial by a binomial similar to or different from long division.
Dividing a polynomial by a binomial is similar to long division in the sense that both processes involve division and finding a quotient. However, there are some key differences.
Here's how you can perform both operations and understand their similarities and differences:
1. Long Division:
- Step 1: Write the dividend (the number being divided) and divisor (the number that divides the dividend) in long division format.
- Step 2: Divide the first digit of the dividend by the divisor. Write the quotient above the division bar.
- Step 3: Multiply the quotient with the divisor and write the result below the dividend.
- Step 4: Subtract the result obtained in step 3 from the portion of the dividend you just considered.
- Step 5: Bring down the next digit from the dividend and repeat steps 2 to 4 until the dividend is entirely used up or until the division process finishes.
2. Polynomial Division by a Binomial:
- Step 1: Arrange the dividend and divisor in descending order of their exponents.
- Step 2: Divide the term with the highest exponent of the dividend by the term with the highest exponent of the divisor. Write the quotient above the division line.
- Step 3: Multiply the entire divisor by the quotient obtained in step 2 and place it below the dividend.
- Step 4: Subtract the result obtained in step 3 from the dividend.
- Step 5: Bring down the next term from the dividend and repeat steps 2 to 4 until all terms of the dividend have been processed or until the division process finishes.
Similarities:
- Both methods involve division and finding a quotient.
- Both methods require subtracting the result of the multiplication from the dividend.
Differences:
- In long division, you deal with numerical digits, whereas in polynomial division by a binomial, you work with variables and their coefficients.
- In polynomial division, you consider terms with ascending exponents, while in long division, you consider digits from left to right.
- In polynomial division, you have variables and exponents, allowing for variable cancellation and simplification.
Overall, while there are similarities between dividing a polynomial by a binomial and long division, they differ in terms of mathematical operations performed and the nature of the values involved.