Problem 29 - Intermediate Algebra

Question

Perform each division. \(\frac{x^{2}-10 x+21}{x-7}\)

Step-by-Step Solution

Answer

The result of the division is \(x - 3\).

Step 1: Set Up the Long Division

To start, set up the division of the polynomial by writing the dividend \(x^2 - 10x + 21\) under the division bracket and the divisor \(x - 7\) outside the bracket. Each step will focus on aligning and subtracting terms.

Step 2: Divide the Leading Terms

Divide the leading term of the polynomial dividend \(x^2\) by the leading term of the divisor \(x - 7\). So, \(x^2 \div x = x\). Place \(x\) above the division bar.

Step 3: Multiply and Subtract

Multiply \(x\) by the divisor \(x - 7\), resulting in \(x^2 - 7x\). Write this below the corresponding terms of the dividend and subtract: \((x^2 - 10x + 21) - (x^2 - 7x) = -3x + 21\).

Step 4: Divide the Resulting Polynomial

Divide the new leading term \(-3x\) by \(x\), which yields \(-3\). Write \(-3\) next to \(x\) above the division bar.

Step 5: Multiply and Subtract Again

Multiply \(-3\) by the divisor \(x - 7\), giving \(-3x + 21\). Subtract this from the current polynomial: \((-3x + 21) - (-3x + 21) = 0\).

Step 6: Conclude the Division

The remainder is zero, indicating that the division process is complete with no remainder left. The result of the division is the expression written above the division bar: \(x - 3\).

Key Concepts

Long Division of Polynomials

Long division of polynomials is a method used to divide one polynomial by another, similar to the long division of numbers. This technique helps simplify complex polynomial expressions and is especially useful when dealing with polynomials of higher degrees. To perform polynomial long division, start by setting up your division as you would with numbers: place the polynomial you want to divide (the dividend) under a long division bracket, and the polynomial by which you want to divide (the divisor) outside the bracket.

Here's the basic process:

• Compare the leading term of your dividend with the leading term of your divisor.
• Divide these terms to find the first term of your quotient.
• Multiply the entire divisor by this first term of the quotient and subtract the result from the dividend or the current polynomial remainder.
• Bring down the next term from the dividend if necessary, and repeat the process until no terms are left to bring down.

This method ensures you systematically break the problem into smaller, manageable steps. Thus, it reveals the quotient and any remainder that is left, providing a thorough understanding of how the division affects each part of the polynomial expression.

Dividing Polynomials by Binomials

Dividing polynomials by binomials often involves the same process as long division, where the divisor is specifically a two-term polynomial called a binomial. This is a common scenario in algebra, especially when simplifying expressions and solving equations.

The process follows the same steps as those outlined for general polynomial long division. Here's a simple approach:

• Identify the dividend: This is the polynomial you want to divide, such as \(x^2 - 10x + 21\).
• Identify the divisor: For example, \(x - 7\).
• Divide the first term of the dividend by the first term of the divisor to find the term of the quotient.
• Multiply the entire binomial (divisor) by this new quotient term, write the result below, and subtract to remove the highest degree term.
• Repeat until the remainder is of lower degree than the divisor.

When dividing by a binomial like \(x - 7\), the division can sometimes result in a remainder of zero, showing that the binomial is a factor of the polynomial. In this case, the division results in an exact quotient, making future algebraic manipulations much more straightforward.

Polynomial Remainder Theorem

The Polynomial Remainder Theorem provides a way to understand and predict remainders in polynomial division without doing the full-blown division process. It states that if you divide a polynomial \(f(x)\) by a linear divisor \(x - k\), the remainder of this division is simply \(f(k)\).

This powerful theorem can simplify many polynomial operations:

• To find the remainder when dividing by \(x - k\), simply evaluate the polynomial at \(x = k\).
• If \(f(k) = 0\), then \(x - k\) is a factor of the polynomial \(f(x)\), making k a root of \(f(x)\).
• It saves time and computational effort, especially in advanced algebra problems, by avoiding repetitive long division operations.

The Remainder Theorem not only uncovers roots and factors conveniently but also offers a quick way to check work after performing long division. Understanding this theorem helps you grasp key concepts in algebra, such as factorization and root finding, more thoroughly and efficiently.