Problem 32 - Algebra 1

Question

Divide. Divide \(c^{2}-25\) by \(c-5\)

Step-by-Step Solution

Answer

The quotient for the division of \(c^{2}-25\) by \(c-5\) is \(c+5\).

Step 1: Perform Polynomial Division

To divide the given expressions, use the formula \(a^{2}-b^{2} = (a-b)(a+b)\). The given expression \(c^{2}-25\) is a difference of squares, so it can be rewritten as: \((c-5)(c+5)\). Now dividing \((c-5)(c+5)\) by \(c-5\) results in \(c+5\). Thus, the division yields \(c+5\).

Step 2: Verify the Result

Test the correctness of the result by performing a multiplication operation with the divisor and quotient. So, multiply \(c+5\) by \(c-5\), which yields \(c^{2}-25\), reflecting the dividend in the original problem.

Key Concepts

Difference of Squares

The difference of squares is a special type of algebraic expression in which two squared terms are subtracted. It follows a specific formula: \(a^2 - b^2 = (a-b)(a+b)\). In this exercise, we have the expression \(c^2 - 25\). Notice how \(25\) is also a perfect square, represented as \((5^2)\). Thus, we can apply the difference of squares formula here.

By recognizing \(c^2 - 25\) as a difference of squares, we can rewrite it as \((c-5)(c+5)\). This form is particularly helpful for simplifying our division task. It breaks down the polynomial into a product of binomials, making it easier for polynomial division. Understanding this concept will allow you to recognize similar patterns in different algebraic problems, simplifying calculations and providing quicker solutions.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operators such as addition, subtraction, multiplication, and division. In polynomial division, the main goal is to simplify these expressions to make them easier to work with.

The given exercise asks us to divide \(c^2 - 25\) by \(c-5\). Recognizing \(c^2 - 25\) as a polynomial, we see it is a binomial expression. This particular combination allows us to simplify it using rules such as the difference of squares. Breaking down such expressions can help us to find simplified solutions, often resulting in a more manageable form like a linear expression – in this case, \(c+5\).

• Identify patterns within polynomials, like difference of squares
• Apply algebraic formulas to simplify expressions
• Break down expressions to find simple and clear results

Verification of Division

Verification is a crucial step in confirming that the division of algebraic expressions has been done correctly. After computing a quotient, it's important to check the accuracy of the result. In algebra, verifying usually involves multiplying the quotient with the divisor to check if it gives the original expression, known as the dividend.

In our exercise, we obtained \(c+5\) as the result of dividing \(c^2 - 25\) by \(c-5\). To verify, multiply \((c+5)\) and \((c-5)\) back together, performing the multiplication:

• Multiply the outer terms: \(c \cdot c = c^2\)
• Multiply the inner terms: \(5 \cdot -5 = -25\)
• Combine these to return to \(c^2 - 25\)

This process of multiplication confirms the division process was correct, reaffirming that the original expression \(c^2 - 25\) matches the result when the quotient and divisor multiply back into the original polynomial. This step safeguards against errors and reinforces understanding of polynomial operations.