Problem 18 - Algebra A Combined Function

Question

Factor each trinomial completely. If a polynomial can't be factored, write "prime." $$ 17+18 n+n^{2} $$

Step-by-Step Solution

Answer

The trinomial factors as \((n+1)(n+17)\).

Step 1: Identify the Quadratic Form

The given trinomial is \(n^2 + 18n + 17\). This expression is in the standard form of a quadratic trinomial \(ax^2 + bx + c\), where \(a = 1\), \(b = 18\), and \(c = 17\).

Step 2: Look for Factor Pairs of c

To factor the trinomial, we need two numbers that multiply to \(c = 17\) and add up to \(b = 18\). Since \(c = 17\) is a prime number, its only factor pairs are \((1, 17)\) and \((-1, -17)\).

Step 3: Check Factor Pair for Sum

Among the factor pairs of 17, the pair \((1, 17)\) adds up to 18, which matches the coefficient of the linear term \(n\). Thus, we can use these numbers for factorization.

Step 4: Write the Factors

Rewrite the trinomial as \((n + 1)(n + 17)\) using the factor pair we identified that sums to \(b = 18\).

Step 5: Confirm the Factorization

Verify the factorization by expanding \((n + 1)(n + 17)\): \((n + 1)(n + 17) = n^2 + 17n + n + 17 = n^2 + 18n + 17\). Since we obtained the original trinomial, our factorization is correct.

Key Concepts

Quadratic Expression

A quadratic expression is a type of polynomial that can be expressed in the standard form as:

• \( ax^2 + bx + c \)

In this format, \( a \), \( b \), and \( c \) are constants, where \( a eq 0 \). The presence of the squared term \( x^2 \) distinguishes quadratics from linear equations.

Quadratics often appear in various mathematical applications, from solving area problems to computing projectile motion in physics. In the context of factoring, recognizing the standard form allows us to systematically apply strategies to simplify the expression into a product of two binomials, if possible.

Prime Number

Prime numbers play a unique role in mathematics. They are integers greater than 1, having exactly two distinct positive divisors: 1 and themselves. For instance, 17 is a prime number because it cannot be divided evenly by any other number except for 1 and 17.

Understanding and identifying prime numbers is crucial when factoring quadratic trinomials, especially when looking for factor pairs of the constant term, \( c \). When \( c \) is a prime number, as in the expression \( n^2 + 18n + 17 \), the factor pairs are limited and often evident, leading to simpler calculations for factorization.

• Factors of 17: \((1, 17)\) and \((-1, -17)\)

Factorization

Factorization in algebra involves breaking down expressions into products of simpler elements, or factors, that when multiplied together give the original expression.

In the context of the given trinomial \( n^2 + 18n + 17 \), factorization follows these key steps:

• Identify pairs of numbers that multiply to the constant term \( c \) and sum to the middle term's coefficient \( b \).
• In this example, since \( c = 17 \) (a prime number), the suitable pair is \( (1, 17) \), as their sum is 18.
• Rewrite the expression as the product of two binomials: \((n + 1)(n + 17)\).

By systematically applying these steps, factorization transforms a quadratic trinomial into a more simplified expression that is easier to analyze or solve.

Polynomials

Polynomials are mathematical expressions composed of variables and coefficients, arranged in terms where each variable is raised to a non-negative integer power. They are a fundamental component in algebra, appearing in numerous forms and complexities.

A trinomial, as seen in the quadratic \( n^2 + 18n + 17 \), is a specific type of polynomial with exactly three terms. Each term contributes to its structure:

• The term \( n^2 \) is the highest degree term and defines it as a quadratic (degree of 2).
• The term \( 18n \) is linear, and its coefficient impacts factorization.
• The constant term 17, being prime, influences how easily the trinomial can be factored.

By understanding these components, one can approach polynomial manipulation, such as factoring or solving, with a clear, systematic strategy that simplifies complex expressions into understandable forms.