Problem 53 - Elementary Algebra

Question

Use a pattern to factor. Check. Identify any prime polynomials. $$ 81 w^{2}+36 w+4 $$

Step-by-Step Solution

Answer

The polynomial \(81w^2 + 36w + 4\) factors to \((9w + 2)^2\). It is not prime.

Step 1: Identify the pattern

First, observe that the quadratic polynomial is in the form of a perfect square trinomial. A perfect square trinomial has the form \(a^2 + 2ab + b^2\). Here, we can try to match the expression \(81w^2 + 36w + 4\) with \(a^2 + 2ab + b^2\).

Step 2: Find a and b

Notice that \(81w^2\) is \((9w)^2\) and \4\ is \2^2\. We need to check if the middle term \36w\ equals \2ab\ when \a = 9w\ and \b = 2\. Calculate: \2 \cdot 9w \cdot 2 = 36w\ which matches the middle term.

Step 3: Express as a perfect square

Since the expression meets the form of a perfect square trinomial, it can be written as \((a + b)^2\), where \a = 9w\ and \b = 2\. Therefore, \(81w^2 + 36w + 4\) factors to \((9w + 2)^2\).

Step 4: Check the factorization

Expand \((9w + 2)^2\) to check if it equals the original expression: \((9w + 2)(9w + 2) = 81w^2 + 18w + 18w + 4 = 81w^2 + 36w + 4\). This confirms that the factorization is correct.

Step 5: Identify as prime or not

A polynomial which can be factored into two or more nontrivial polynomials is not prime. Since \(81w^2 + 36w + 4\) can be expressed as \((9w + 2)^2\), it is not prime.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a specific type of quadratic polynomial that takes the form \(a^2 + 2ab + b^2\). This structure allows it to be easily factored into \( (a + b)^2 \).
Recognizing these trinomials can make factoring much simpler since you know the resulting binomial will be squared.
Take, for example, the expression 81w² + 36w + 4.

• First, identify that 81w² can be written as (9w)² and 4 can be written as 2².
• Then, check if the middle term, 36w, matches the form 2ab. Here, a=9w and b=2.
• Calculating: 2 * 9w * 2 = 36w, confirms our middle term fits the pattern.

This confirms that 81w² + 36w + 4 can be factored as (9w + 2)², making it a perfect square trinomial.

Factoring Technique

Factoring is a method used to break down a polynomial into simpler components called factors, which when multiplied give back the original polynomial.
Different techniques can be used depending on the form of the polynomial.

• For example, when you encounter a quadratic polynomial like 81w² + 36w + 4, you need to look for patterns that could simplify the factoring process.
• One common pattern is the perfect square trinomial.

By identifying the structure of 81w² + 36w + 4, we quickly see it's a perfect square trinomial and can use the factoring technique for such forms. Rewriting the polynomial as \( (9w + 2)^2 \) simplifies our work. Always double-check your factored form by expanding it to ensure it matches the original polynomial.

Prime Polynomials

A prime polynomial is a polynomial that cannot be factored further into simpler polynomials with integer coefficients. It's akin to prime numbers in arithmetic.
Consider the polynomial 81w² + 36w + 4. Since we managed to factor it into \( (9w + 2)^2 \), it is not a prime polynomial. Prime polynomials are those that remain unfactored, except by 1 and the polynomial itself. If a polynomial can be broken down into more manageable sub-polynomials, it is considered composite, not prime.
For a polynomial to be truly prime, no matter how you probe or test with different factoring techniques, it won't split into integer factors beyond itself and one.

Polynomial Factorization

Polynomial factorization involves breaking down polynomials into simpler, multipliable components, thereby simplifying operations like solving equations or further mathematical processes.
To factorize a polynomial like 81w² + 36w + 4:

• Identify any recognizable patterns, like perfect square trinomials.
• Express the polynomial in its factorized form.

Here we observed that 81w² + 36w + 4 fits the form \(a^2 + 2ab + b^2\), allowing us to factor it efficiently as \( (9w + 2)^2 \). This process of systematically breaking down and checking ensures that factorization is accurate and that the polynomial is expressed in its simplest form for further use in algebraic tasks.