Problem 63 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{20 y^{2}+15 y-4}{4 y+3} $$

Step-by-Step Solution

Answer

Answer: The simplified expression is \( 5y - \frac{4}{4y+3} \).

Step 1: Set up the long division

First, set up the long division with the polynomial in the numerator and the monomial in the denominator. Write the division symbol and place the numerator, \(20y^2 + 15y - 4\), inside it and the denominator, \(4y + 3\), outside it.

Step 2: Divide the first term

Divide the first term of the numerator, \(20y^2\), by the first term of the denominator, \(4y\). This gives \(5y\). Write this quotient on top of the division symbol.

Step 3: Multiply and subtract

Now, multiply the quotient, \(5y\), by the denominator, \(4y + 3\). This gives \(20y^2 +15y\). Then, subtract this product from the initial numerator: $$ \begin{array}{c|cc cc} \multicolumn{2}{r}{5y} \\ \cline{2-5} 4y+3 & 20y^2 & +15y & -4 \\ \cline{2-3} & 20y^2 & +15y \\ \cline{2-3} & & & -4 \\ \end{array} $$

Step 4: Final expression

There are no more terms to subtract in the numerator, so the final result of the long division is the quotient, \(5y\), with a remainder of \(-4\). The remainder should also be divided by the denominator, so the final expression is: $$ 5y - \frac{4}{4y+3}. $$

Key Concepts

Long Division in Algebra

Long division in algebra is a process used to divide polynomials, just like you would do with numbers. The main goal is to simplify the given expression neatly. It might seem daunting at first, but breaking it down into smaller steps makes it much easier. When performing long division:

• Start by identifying both parts of the fraction: the numerator and the denominator.
• Place the polynomial (numerator) beneath the long division bracket and the divisor (denominator) outside.
• Divide the leading term of the numerator by the leading term of the denominator. Write this quotient above the division line.

The process continues by multiplying back this quotient with the divisor and subtracting the result from your current polynomial. Repeat these steps until every term is accounted for, either by becoming part of the quotient or residue. The long division ends when the remaining polynomial's degree is less than that of the divisor. Any leftover expressions are written as a remainder fraction attached to the quotient, giving you a final answer.

Rational Expressions

Rational expressions are fractions that have polynomials in both the numerator and the denominator. Just like with regular fractions, you can perform operations like addition, subtraction, multiplication, and division. Simplifying rational expressions involves reducing the polynomial fraction to its simplest form. This is done by detecting any common factors between the numerator and the denominator and canceling them out. Ensure that the denominator isn't zero, as division by zero is undefined in mathematics. When dividing rational expressions, it's often helpful:

• To factor both the numerator and the denominator.
• Cancel any common factors they might share.
• Proceed with the division using techniques like polynomial long division.

Being comfortable with rational expressions provides a foundation for more complex algebraic operations. Understanding them equips you with tools to succeed in higher-level math.

Polynomial Arithmetic

Polynomial arithmetic involves performing basic operations like addition, subtraction, multiplication, and division on polynomials. It requires understanding the rules of combining like terms and managing coefficients, which is like keeping track of the numbers in front of variables. Addition and subtraction of polynomials are straightforward:

• Align and combine like terms, which are terms with the same variable and exponent.

Multiplication of polynomials involves applying the distributive property:

• Each term in one polynomial gets multiplied with every term in the other.
• Combine like terms to simplify.

In division, as showcased in the original exercise, long division is typically used. This process allows the breakdown of a larger polynomial into simpler parts or even finding roots if entirely divisible. Polynomial arithmetic is foundational to understanding algebraic structures and functions, making it beneficial to master these operations early on.