Problem 14 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

(2y + 9z)^2

Step 1: Recognize the Form of the Polynomial

The given polynomial is a quadratic trinomial of the form \(4y^2 + 36yz + 81z^2\). Notice that this form looks similar to the square of a binomial.

Step 2: Write Down the Binomial Squares

Recognize that \(4y^2\) is the same as \((2y)^2\) and \(81z^2\) is the same as \((9z)^2\).

Step 3: Identify the Middle Term

Check if the middle term \(36yz\) matches \(2ab\) where \(a = 2y\) and \(b = 9z\). Calculate \(2(2y)(9z) = 36yz\). The middle term matches.

Step 4: Write the Factored Form of the Polynomial

Since the middle term matches, the polynomial can be factored as the square of a binomial: \((2y + 9z)^2\).

Step 5: Check the Factoring

Expand \((2y + 9z)^2\) to verify: \((2y + 9z)(2y + 9z) = 4y^2 + 36yz + 81z^2\). This confirms the original polynomial was correctly factored.

Step 6: Identify Any Prime Polynomials

There are no further factors since \((2y + 9z)\) is a binomial and cannot be factored further. Hence, \(4y^2 + 36yz + 81z^2\) has no prime polynomial factors.

Key Concepts

quadratic trinomials

Quadratic trinomials are polynomials of degree two, meaning the highest exponent of the variable is 2. They are generally written in the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our example, the quadratic trinomial is \(4y^2 + 36yz + 81z^2\). To factor these trinomials, we often look for patterns or use specific techniques such as completing the square or using the quadratic formula. Factoring can simplify solving the equation or make it easier to work with in subsequent problems. Recognizing the form of the polynomial is the crucial first step. Always ensure that you look for common factors first, and then identify any notable patterns.

binomial squares

Binomial squares are specific forms of expressions that result from squaring a binomial. A binomial is simply a polynomial with two terms, such as \(a + b\) or \(a - b\). When you square these, you get expressions like \((a + b)^2\) or \((a - b)^2\). For example, squaring \((2y + 9z)\) gives us \((2y + 9z)^2 = 4y^2 + 36yz + 81z^2\). The binomial square pattern is:

• \((a + b)^2 = a^2 + 2ab + b^2\)

Recognizing this pattern allows us to quickly match middle terms and identify that we can represent our quadratic trinomial as the square of a binomial. Consistently check middle terms to see if they match the form \(2ab\) to validate that it is indeed a binomial square.

prime polynomials

Prime polynomials are polynomials that cannot be factored further over a given set of numbers, typically the integers. They are the 'building blocks' of polynomials, similar to prime numbers in integers. To determine if a polynomial is prime, try factoring it. If it cannot be broken down into simpler polynomials with integer coefficients, it is prime.
In our case, after factoring \(4y^2 + 36yz + 81z^2\) as \((2y + 9z)^2\), we check if \(2y + 9z\) can be factored further. Since \(2y + 9z\) is already a binomial with no common factors, it cannot be simplified further. This confirms that \(4y^2 + 36yz + 81z^2\) has no prime polynomial factors other than itself and cannot be factored into simpler prime polynomials over the integers.