Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=x^{5}-1 ; x-1$$

Step-by-Step Solution

Answer

The quotient is \(x^4 + x^3 + x^2 + x + 1\), with a remainder of 0.

Step 1: Understanding Polynomial Long Division

Polynomial long division is a method used to divide polynomials. Here, we need to divide \(P(x) = x^5 - 1\) by the binomial \(x - 1\). The process is similar to numerical long division.

Step 2: Setup the Division

Write the polynomial \(P(x) = x^5 - 1\) under the division bar, and \(x - 1\) outside the division bar. Make sure to align terms of the same power together.

Step 3: Divide the First Term

Divide the first term of the dividend \(x^5\) by the first term of the divisor \(x\), which gives you \(x^4\). Write \(x^4\) above the division bar.

Step 4: Multiply and Subtract

Multiply \(x^4\) by \(x - 1\), yielding \(x^5 - x^4\). Subtract this result from \(x^5 - 1\), which simplifies to \(x^4 - 1\).

Step 5: Repeat the Process

Take the new polynomial \(x^4 - 1\) as the new dividend. Repeat the division process for the term \(x^4\) (divide, multiply, subtract) until you reach a remainder with a lower order than the divisor.

Step 6: Perform Remaining Divisions

Continue the division until all terms have been dealt with:- Divide \(x^4\) by \(x\) to get \(x^3\), continue with subtraction similarly.- Divide the resulting first term by \(x\) until you reduce the dividend lower than the divisor.

Step 7: Determine the Quotient and Remainder

After performing each division step, the quotient comes out to \(x^4 + x^3 + x^2 + x + 1\) and the remainder is 0, meaning \(x^5 - 1 = (x - 1)(x^4 + x^3 + x^2 + x + 1)\).

Key Concepts

Polynomial Division

Polynomial division works similarly to long division with numbers. It is used when you need to divide a polynomial by another polynomial, typically resulting in a quotient and possibly a remainder. This method is especially useful when the divisor is a binomial, making it an approachable process.

• Begin by setting up the division, placing the dividend (the polynomial you are dividing) under a division bar and the divisor (the polynomial you are dividing by) outside of it.
• The leading terms, which are the terms with the highest powers in both polynomials, are divided first.
• Multiply the result by the entire divisor and subtract from the dividend or the resulting polynomial.
• Carry down the next term of the dividend and repeat the process.

This cycle continues until all terms of the original polynomial have been divided, and any remaining polynomial has a lower degree than the divisor.

Quotient and Remainder

The result of polynomial division is expressed in terms of a quotient and a remainder, similar to division with integers.
- **Quotient**: This is the result obtained from the division of the polynomial. In our original exercise, after dividing the polynomial \(P(x) = x^5 - 1\) by \(x - 1\), the quotient obtained is \(x^4 + x^3 + x^2 + x + 1\). This quotient represents the full value apart from the remainder.
- **Remainder**: This is the part of the dividend that is left after division and has a degree less than the divisor. In polynomial division, the remainder must have a degree lower than that of the divisor. For our example, the remainder is 0, which means the division was exact.
The relationship can be denoted by the equation: \[P(x) = (x - 1)(x^4 + x^3 + x^2 + x + 1) + 0\] which confirms the quotient and remainder calculation.

Binomial Division

Binomial division is a specific case of polynomial division where you are dividing by a binomial, a polynomial of degree one. It’s straightforward because you only have to deal with single terms.
Using the exercise as an example, the binomial \(x - 1\) is what we divide the polynomial \(P(x) = x^5 - 1\) by.
The key steps include:

• Focusing on dividing the highest degree term of the dividend by the leading term of the divisor, in this case: \(x^5\) divided by \(x\) resulting in \(x^4\).
• This process gives us part of the quotient, which we then use to subtract the product of the quotient and binomial from the original polynomial.
• Each subtraction step reduces the dividend degree until the remainder is of a smaller degree than the binomial.

The systematic approach of binomial division ensures accuracy and simplifies handling powers step-by-step.

Polynomial Expressions

Polynomial expressions consist of variables raised to non-negative integer exponents, typically written in descending order of these powers. In \(P(x) = x^5 - 1\), \(x^5\) is the term with the highest degree, indicating the shape and end behavior of the graph of the polynomial.

• The degree of a polynomial is the highest power of the variable, and it influences the polynomial division. In our case,\x^5\ is the highest degree term.
• When performing polynomial division, each term is divided sequentially, starting with the highest degree down to the lowest, ensuring clarity and order in the calculation process.
• Polynomials can also be simplified after division, as seen from obtaining the quotient \(x^4 + x^3 + x^2 + x + 1\) that neatly approaches the whole value of \(P(x)\) without a remainder when divided by \(x - 1\).

Understanding the structure and order of polynomial expressions is vital to mastering division and simplifying complex algebraic calculations.