Introduction

In algebra, dividing standard form is a fundamental operation that involves dividing a polynomial in standard form by another polynomial. Standard form refers to a polynomial arranged in descending powers of the variable, with the coefficients of each term written explicitly. Dividing standard form allows us to simplify complex expressions and solve various algebraic problems.

To divide standard form effectively, we must understand the concept of long division in polynomial form. Long division is an algorithm that breaks down a complex division problem into a series of simpler steps, ultimately yielding the quotient and remainder of the division.

This article will provide a comprehensive guide to dividing standard form, explaining each step of the process in detail. We will cover various scenarios and provide numerous examples to solidify your understanding.

Setting Up the Division Problem

Step 1: Divide the First Terms

Align the two polynomials vertically with the divisor (the polynomial you are dividing by) on the right. Divide the first term of the dividend (the polynomial you are dividing) by the first term of the divisor, and write the result above the dividend.

Example:

(x^3 - 2x^2 + x + 1) ÷ (x - 1)
  x^2 - x + 1

Step 2: Multiply the Divisor by the Quotient

Multiply the divisor by the quotient you obtained in step 1. Write the result below the dividend, aligning it vertically.

Example:

(x^3 - 2x^2 + x + 1) ÷ (x - 1)
  x^2 - x + 1
  ------------
  x^3 - x^2 + x - x^2

Step 3: Subtract the Products

Subtract the result from step 2 from the dividend. The difference will become the new dividend.

Example:

(x^3 - 2x^2 + x + 1) ÷ (x - 1)
  x^2 - x + 1
  ------------
  x^3 - x^2 + x - x^2
  ----------------
  -x^2 + x + 1

Bringing Down the Next Term

Bring down the next term of the dividend next to the new dividend. This will form the new dividend for the next step.

Example:

(x^3 - 2x^2 + x + 1) ÷ (x - 1)
  x^2 - x + 1
  ------------
  x^3 - x^2 + x - x^2
  ----------------
  -x^2 + x + 1
  -x^2 + x
  ----------------

Repeating the Steps

Step 4: Divide the First Terms (Again)

Repeat the process from step 1: divide the first term of the new dividend by the first term of the divisor. Write the result above the new dividend.

Example:

(x^3 - 2x^2 + x + 1) ÷ (x - 1)
  x^2 - x + 1
  ------------
  x^3 - x^2 + x - x^2
  ----------------
  -x^2 + x + 1
  -x^2 + x
  ----------------
  +1

Step 5: Multiply the Divisor by the Quotient (Again)

Repeat the process from step 2: multiply the divisor by the quotient you obtained in step 4. Write the result below the new dividend, aligning it vertically.

Example:

(x^3 - 2x^2 + x + 1) ÷ (x - 1)
  x^2 - x + 1
  ------------
  x^3 - x^2 + x - x^2
  ----------------
  -x^2 + x + 1
  -x^2 + x
  ----------------
  +1
  +1

Step 6: Subtract the Products (Again)

Repeat the process from step 3: subtract the result from step 5 from the new dividend. This will become the new dividend.

Example:

(x^3 - 2x^2 + x + 1) ÷ (x - 1)
  x^2 - x + 1
  ------------
  x^3 - x^2 + x - x^2
  ----------------
  -x^2 + x + 1
  -x^2 + x
  ----------------
  +1
  +1
  ----------------
  0

Interpreting the Result

Once the final division step results in a remainder of zero, the division is complete. The quotient obtained represents the polynomial that, when multiplied by the divisor, yields the dividend. The remainder, if any, represents the portion of the dividend that cannot be divided evenly by the divisor.

In the example above, the quotient is x^2 – x + 1 and the remainder is 0. This means that:

(x^3 - 2x^2 + x + 1) ÷ (x - 1) = (x^2 - x + 1)

Dividing by Polynomials of Degree Greater Than 1

The process of dividing standard form is similar for polynomials of degree greater than 1. The key is to follow the same steps outlined above, starting with the highest degree term of both the dividend and divisor.

Example:

(x^4 + 2x^3 - 5x^2 + 7x - 1) ÷ (x^2 - 2x + 1)
  x^2 + 4x - 3
  ------------
  x^4 + 2x^3 - 5x^2 + 7x - 1
  x^4 - 2x^3 + x^2
  ----------------
  4x^3 - 6x^2 + 7x - 1
  4x^3 - 8x^2 + 4x
  ----------------
  2x^2 + 3x - 1
  2x^2 - 4x + 2
  ----------------
  7x - 3

Special Cases

Dividing by a Term

If the divisor is a term (a polynomial with only one term), the division process is simplified.

Example:

(x^3 + 2x^2 - 5x + 7) ÷ x
  x^2 + 2x - 5 + 7/x

Dividing by a Constant

If the divisor is a constant, the division process involves dividing each coefficient of the dividend by the constant.

Example:

(x^3 + 2x^2 - 5x + 7) ÷ 2
  (1/2)x^3 + x^2 - (5/2)x + 7/2

Dividing by a Binomial

When dividing standard form by a binomial, it is useful to factor the divisor and apply the difference of squares formulas. This can simplify the division process.

Example:

(x^2 - 4) ÷ (x + 2)
  x - 2
  ------------
  x^2 - 4
  x^2 + 2x
  ----------------
  -2x - 4
  -2x - 4
  ----------------
  0

Conclusion

Dividing standard form is a fundamental operation in algebra that allows us to simplify complex expressions and solve various problems. By understanding the steps involved in long division in polynomial form, you can effectively divide polynomials and deepen your knowledge of algebra.

Remember to practice regularly and apply the techniques outlined in this guide to improve your proficiency in dividing standard form. With consistent practice, you will become confident in performing this operation and expand your algebraic skills.

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