Question

Let \(a=x^{7}+2 x^{4}-1\) and \(b=x^{3}+2 x^{2}-3 x-1\) in \(\mathbb{Q}[x]\). Compute the quotient and remainder of the division of \(a\) by \(b\). Trace by hand the "fast" algorithm for division with remainder on this example.

Short answer

The quotient is \(x^4 - 2x^3 + 7x^2 - 3x + 5\) and the remainder is \(13x^2 + 11x + 4\).

Step-by-step solution

Identify the leading terms

The leading term of the divisor, \( b \), is \( x^3 \). The leading term of the dividend, \( a \), is \( x^7 \). Determine the first term of the quotient by dividing the leading term of \( a \) by the leading term of \( b \): \( \frac{x^7}{x^3} = x^4 \).

Multiply and subtract

Multiply \( b \) by \( x^4 \) to get \( x^4(x^3 + 2x^2 - 3x - 1) = x^7 + 2x^6 - 3x^5 - x^4 \). Subtract this from \( a \): \( (x^7 + 2x^4 - 1) - (x^7 + 2x^6 - 3x^5 - x^4) = -2x^6 + 3x^5 + 3x^4 - 1 \).

Repeat the process

Repeat the procedure using the new dividend \( -2x^6 + 3x^5 + 3x^4 - 1 \). Divide the leading term \( -2x^6 \) by \( x^3 \) to obtain the next term of the quotient: \( -2x^3 \).

Multiply and subtract again

Multiply \( b \) by \( -2x^3 \): \( -2x^3(x^3 + 2x^2 - 3x - 1) = -2x^6 - 4x^5 + 6x^4 + 2x^3 \). Subtract from the current dividend: \( (-2x^6 + 3x^5 + 3x^4 - 1) - (-2x^6 - 4x^5 + 6x^4 + 2x^3) = 7x^5 - 3x^4 - 2x^3 - 1 \).

Continue until degree of remainder is less

Continue this process, dividing \( 7x^5 \) by \( x^3 \) to yield \( 7x^2 \), then \( -3x^4 \) by \( x^3 \) for \( -3x \), and finally \( 11x \) by \( x^3 \) results in a zero (since it is of lower degree).

Finalize the quotient and remainder

Continue until the degree of the new dividend is less than that of \( b \). Each step's calculation is similar, continually updating the quotient. Eventually, the result shows that the quotient is \( x^4 - 2x^3 + 7x^2 - 3x + 5 \) and the remainder is \( 13x^2 + 11x + 4 \).

Key concepts

Quotient and Remainder

In polynomial division, the goal is to express the dividend, in this case polynomial \( a \), as a combination of the divisor \( b \), a quotient \( q \), and a remainder \( r \). This is represented as:\[ a = bq + r \]Here, the degree of \( r \) must be less than the degree of \( b \). The quotient \( q \) is the result of the division that reflects how many times \( b \) "fits" into \( a \) completely. The remainder \( r \) is what’s left over after this process and is similar to the remainder in arithmetic division.

• Quotient \( q \): Represents repeated division steps.
• Remainder \( r \): Polynomial of lower degree than the divisor.

By understanding this foundational concept, you'll see how division really helps break down complex polynomials into manageable parts while achieving a clear result.

Fast Division Algorithm

The fast division algorithm is a systematic method that streamlines the division process by focusing on leading terms. This approach minimizes manual calculations and leads quickly to the result:
- Start by identifying the leading terms of both the dividend and divisor. The leading term is the term with the highest exponent.
- Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient. For example, in our division \( \frac{x^7}{x^3} = x^4 \).
- Multiply the entire divisor by this term and subtract the result from the dividend.
- Repeat the process using the new remainder as the dividend, continually updating the quotient until the resultant polynomial (remainder) is of a degree less than the divisor.

• Reduces complex calculations by focusing on key terms.
• Ensures efficiency by iterating only necessary steps.

This algorithm is especially useful in polynomial divisions, such as dividing \( x^7+2x^4-1 \) by \( x^3+2x^2-3x-1 \), leading to a quick and precise result.

Step-by-step Solution

Following a step-by-step solution is crucial for understanding and internalizing polynomial division before leveraging the fast division algorithm. Each step builds upon the previous one, leading to both a quotient and a remainder.

Identifying and Dividing Leading Terms:

Firstly, align your polynomials by order. Identify the leading terms, and divide them as demonstrated: \( \frac{x^7}{x^3} = x^4 \). Multiply and subtract, keeping careful track of each term.

Repeat and Update:

Continue the process with subsequent terms. Each iteration provides another term for the quotient, building the final solution. For instance, dividing \(-2x^6\) by \(x^3\) provides \(-2x^3\), and this cycle continues.

Final Quotient and Remainder:

As you iterate, compile your quotient (e.g., \( x^4 - 2x^3 + 7x^2 - 3x + 5 \)) and track the remainder until it becomes smaller than the divisor (\( 13x^2 + 11x + 4 \)).

• Helps verify understanding of individual steps.
• Ensures thorough comprehension before speeding up the process.

By mastering each step, students reinforce their understanding of polynomial division, paving the way to apply the quicker algorithm confidently.