Problem 14 - Elementary Algebra

Question

Use the guess and check method to factor. Identify any prime polynomials. $$ y^{2}+16 y+64 $$

Step-by-Step Solution

Answer

The polynomial factors to \[\begin{equation} (y + 8)^2 \text { and is not prime.}

Step 1 - Write the polynomial in standard form

The given polynomial is already in standard form: y^{2} + 16y + 64

Step 2 - Identify the coefficients

Identify the coefficients: a = 1 b = 16 c = 64

Step 3 - Guess potential factors of the constant term

Check for pairs of numbers that multiply to the constant term, 64. Possible pairs include: (1, 64), (2, 32), (4, 16), (8, 8)

Step 4 - Check pair whose sum is the middle coefficient

Identify if any of these pairs sum to 16 (the coefficient of y). The pair (8, 8) works since 8 + 8 = 16.

Step 5 - Factor the polynomial

Since (8, 8) are the numbers, the polynomial can be factored into: (y + 8)(y + 8)

Step 6 - Write the final factored form

The factored form is: (y + 8)^2

Step 7 - Identify if the polynomial is prime

Since the polynomial can be factored, it is not a prime polynomial.

Key Concepts

Guess and Check Method

The guess and check method is a straightforward approach to factor polynomials. It involves guessing pairs of numbers that could potentially be the factors of the polynomial and then checking if these pairs work. This method is particularly useful when dealing with simple polynomials.

To use the guess and check method:

• Identify the coefficients in the polynomial. For example, in the polynomial \(y^2 + 16y + 64\), the coefficients are: \(a = 1\), \(b = 16\), and \(c = 64\).
• Guess pairs of numbers that when multiplied give the constant term, \(c\). For instance, possible pairs for 64 are: \((1, 64)\), \((2, 32)\), \((4, 16)\), and \((8, 8)\).
• Check which pair, when added, equals the middle coefficient \(b\). In this case, the pair \((8, 8)\) works because \(8 + 8 = 16\).

Once you find the correct pair, you can write the factors of the polynomial. For example, \(y^2 + 16y + 64\) factors into \((y + 8)(y + 8)\), which can be simplified to \((y + 8)^2\). By breaking it down this way, the guess and check method makes factoring easier to understand and apply.

Prime Polynomials

Prime polynomials are polynomials that cannot be factored further over the set of integers. This means that there are no two non-constant polynomials that can be multiplied to produce the original polynomial.

To determine if a polynomial is prime:

• First, attempt to factor the polynomial using methods such as the guess and check method, grouping, or the quadratic formula.
• If the polynomial cannot be factored by any of these methods, it is likely prime.
• For example, the polynomial \(y^2 + 16y + 64\) is not prime since it can be factored into \((y + 8)^2\).

Prime polynomials have special significance because they are the building blocks for all polynomials, much like prime numbers are for integers. Understanding whether a polynomial is prime helps in solving more complex algebra problems.

Standard Form

A polynomial is in standard form when its terms are written in descending order of their degrees. This makes it easier to work with and identify the key components of the polynomial.

For example, the polynomial \(y^2 + 16y + 64\) is in standard form because the terms are listed from the highest degree to the lowest:

• \(y^2\), which has a degree of 2
• \(16y\), which has a degree of 1
• \(64\), the constant term, with a degree of 0

When a polynomial is in standard form, you can easily identify the coefficients for the guess and check method or any other factoring methods. Correctly identifying these coefficients is a critical first step in successfully solving polynomial equations.