Problem 38 - Algebra 1

Question

Divide. Divide \(5-7 m+3 m^{2}\) by \(m-3\)

Step-by-Step Solution

Answer

The solution to \(5-7 m+3 m^{2}\) divided by \(m-3\) is \(3m + 2 - \frac{1}{m-3}\)

Step 1: Set Up the Long Division

When setting up the division, write the terms of both the dividend and divisor in descending order of exponents. In this case, \(3 m^{2} - 7 m + 5\) is our dividend and \(m - 3\) is our divisor.

Step 2: Divide the Leading Terms

The first step in polynomial division is dividing the first term of the dividend by the first term of the divisor. So, \(3m^{2} \div m = 3m\). This is the first term of our quotient.

Step 3: Multiply and Subtract

Next, multiply the divisor by the first term of the quotient, then subtract this result from the original polynomial. \((3m)(m - 3) = 3m^{2} - 9m\). Subtracting this from our original dividend yields a new polynomial, \(2 m + 5\).

Step 4: Repeat the Divide, Multiply and Subtract

Repeat the process of dividing the first term of what's left from the last step by the first term of the divisor, \(2m \div m = 2\), our next quotient term. Multiply the divisor by this quotient term and subtract from the remainder to get our final remainder, \(5 - 2*3 = -1\).

Step 5: Write down the final quotient and remainder

The solution to our division is the quotient plus the remainder divided by our divisor. In this case, that's \(3m + 2 - \frac{1}{m-3}\).

Key Concepts

Dividing Polynomials

Understanding how to divide polynomials is essential for solving a variety of algebraic problems. Much like numerical long division, polynomial division is a method used to divide a polynomial (the dividend) by another polynomial (the divisor), which simplifies the expression into a quotient and possibly a remainder.

When dealing with polynomial long division, it's important to arrange both the dividend and divisor in descending power. Then, much like long division with numbers, you divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient. This process continues iteratively until you’re left with a term that can no longer be divided by the divisor, resulting in a remainder.

Polynomial Division Steps

Polynomial division follows a specific sequence of steps that, if executed correctly, will yield the quotient and remainder. Here are the steps explained with our example, dividing \(3m^2 - 7m + 5\) by \(m - 3\):

Initial Setup

Write out the dividend and divisor in descending order of their exponents. This setup is crucial for keeping track of the terms through each step of the division.

Divide Leading Terms

Divide the first term of the dividend by the first term of the divisor. The result is the first term of the quotient.

Multiply and Subtract

Multiply the entire divisor by the new term of the quotient, and subtract this product from the dividend to get a new polynomial.

Repeat the Process

Continue the process with the new polynomial, dividing the leading term by the leading term of the divisor. Subtract again to obtain another new polynomial, and repeat until the polynomial that remains cannot be divided by the divisor.

Write the Final Answer

The quotient and the remainder make up the final answer to the polynomial division problem. If the remainder is non-zero, it is written as a fraction over the original divisor in the final expression.

Descartes' Rule of Signs

Descartes' Rule of Signs is a valuable theorem in algebra for determining the number of positive and negative real roots in a polynomial equation. While it's not directly used in polynomial long division, it can provide insights into the nature of the polynomial factors and roots prior to division.

The rule states that the number of positive real roots of a polynomial equation equals the number of sign changes in the coefficients of the terms when arranged in descending order of their degrees, or is less than that by an even number. Similarly, to find the number of negative real roots, substitute the variable with its negative counterpart and then apply the rule. Keep in mind, this rule counts only real roots, not complex ones, and it gives possible root counts rather than exact amounts.

Understanding this rule can provide an advantage when interpreting the results of polynomial long division, particularly if further factorization of the quotient is needed or if you're analyzing the roots of the divisor.