Problem 11 - Beginning and Intermediate Algebra

Question

Complete the factorization. $$c^{2}-14 c+45=(c-5)(\quad)$$

Step-by-Step Solution

Answer

The complete factorization is \(c^2 - 14c + 45 = (c - 5)(c - 9)\).

Step 1 : Find the First Term

We need to find a term that, when multiplied by the first term of the given expression (\(c\)), will result in the first term of the quadratic expression (\(c^2\)). Since there's only one variable, that term must be \(c\). Now we have \((c-5)(c + \ ?)\), and we need to find the second term.

Step 2: Find the Second Term

We need to find a value that, when multiplied by \(-5\), will result in the constant term of the quadratic expression, which is \(45\). Since \(-5 * -9 = 45\), the second term must be \(-9\).

Step 3: Write the Complete Factorization

Now that we've found both terms, we can write the complete factorization as: \[c^2 - 14c + 45 = (c - 5)(c - 9)\].

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical expressions that consist of variables, coefficients, and exponents. The degrees of the terms provide a clear structure of how complex the expression is.
In our specific problem, the polynomial expression is a quadratic expression, which is a polynomial of degree 2.
Quadratic expressions have three terms, represented in the general form of \(ax^2 + bx + c\). Each term has a specific role:

• The first term involves the variable squared (in our case, \(c^2\)).
• The second term involves the variable multiplied by a coefficient (\(-14c\)).
• The third term is a constant (\(45\)).

Understanding the structure of these expressions is critical for tasks like factorization, which is breaking down the expression into simpler multiplier factors.

Quadratic Equations

Quadratic equations are equations in which the highest power of the variable is 2. They are often seen in the form of \(ax^2 + bx + c = 0\). In our exercise,
we started with the quadratic expression \(c^2 - 14c + 45\). To solve or factor a quadratic equation involves looking for two numbers that multiply to make the constant term, while adding together to make the linear coefficient (-14 in our case).
In our exercise, the factorization asked for us to fill in the missing term in \((c-5)(c+?)\).
This process involves:

• Identifying the first term is always straightforward as it leads to \(c\).
• Finding such pairs of numbers (in this case, \(c-9\) with the condition that their product is the constant term \(45\)).

Using these numbers, we can break down the quadratic expression into the factors \((c - 5)(c - 9)\), which represents two instances where the equation product equals zero: \(c = 5\) and \(c = 9\).

Algebraic Expressions

Algebraic expressions form the core of algebra, comprised of numbers, variables, and arithmetic operations. They are simplified or factored, based on what the problem demands.
In our case, simplifying involves factorization, which requires breaking a quadratic equation down into simpler expressions. Through the step-by-step process in our original solution, we identified the missing factor in the expression,
thereby simplifying the algebraic expression into two binomials.
The steps to handle algebraic expressions involve:

• Recognizing the type of polynomial (quadratic, linear, cubic, etc.).
• Understanding relationships between coefficients and constants (like finding terms that multiply to the constant and add up to the linear coefficient).
• Applying algebraic identities and factorization techniques to rewrite the expression in a factored form.

This powerful tool set allows us to transform expressions and better understand mathematical relationships, ensuring clarity when solving problems.