Problem 17 - Elementary Algebra

Question

Use a pattern to factor. Check. Identify any prime polynomials. $$ y^{2}-6 y z+9 z^{2} $$

Step-by-Step Solution

Answer

The factorized form is $$(y-3z)^2$$ and it is not a prime polynomial.

Step 1: Recognize the Pattern

Identify the given polynomial and see if it matches any special factoring patterns such as a perfect square trinomial or difference of squares. The given polynomial is $$y^{2}-6 y z+9 z^{2}$$ which seems to fit the pattern of a perfect square trinomial.

Step 2: Identify the Squares

Determine if the first and the last terms of the polynomial are perfect squares. Here, the first term is $$y^2 = (y)^2$$ and the last term is $$9z^2 = (3z)^2$$.

Step 3: Verify the Middle Term

Check if the middle term follows the pattern $$2ab$$, where $$a = y$$ and $$b = 3z$$. We have $$2 \times y \times 3z = 6yz$$, and since our middle term is $$-6yz$$, it fits the pattern with a negative sign.

Step 4: Write the Factorized Form

Since the polynomial fits a perfect square trinomial pattern, it can be written as: $$ (a - b)^2 $$. Replace $$a$$ and $$b$$ with the identified terms: $$ y^{2} - 6 y z + 9 z^{2} = (y - 3z)^2 $$

Step 5: Check if Factorized Form is Correct

Expand $$(y-3z)^2$$ to verify correctness: $$ (y-3z)(y-3z) = y^2 - 3yz - 3yz + 9z^2 = y^2 - 6yz + 9z^2 $$ which matches the original polynomial.

Step 6: Identify Any Prime Polynomials

In this case, the polynomial $$y^2 - 6yz + 9z^2$$ is not a prime polynomial since it can be factored into $$(y-3z)^2$$.

Key Concepts

Perfect Square Trinomial

In algebra, a perfect square trinomial is one of the most recognized patterns. A perfect square trinomial is a quadratic expression that can be rewritten as the square of a binomial. For example, the expression \(y^2 - 6yz + 9z^2\) is a perfect square trinomial. Let's break down why:

The general form of a perfect square trinomial is \(a^2 - 2ab + b^2\). Here, it transforms into \((a - b)^2\).
Identify the squares:

• The first term, \(y^2\), is a perfect square as \(y^2 = (y)^2\).
• The last term, \(9z^2\), is also a perfect square since \(9z^2 = (3z)^2\).

Next, look at the middle term: It follows the pattern of \(-2ab\) where \(2ab = 2 \times y \times 3z\)

• Here \(a = y\) and \(b = 3z\). So \(2ab = 2 \times y \times 3z = 6yz\)

The original polynomial has \(-6yz\) as its middle term, with the negative sign consistent. Hence, we can write \(y^2 - 6yz + 9z^2\) as \((y-3z)^2\).

Checking for correctness, expanding \((y-3z)^2\) returns \(y^2 - 6yz + 9z^2\), validating our factored form.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler 'factor' polynomials whose product is the original polynomial. This makes solving equations easier and simplifies expressions.

The given polynomial \(y^2 - 6yz + 9z^2\) can be factorized because it is a perfect square trinomial.
Here's how to factorize it:

• Identify any patterns, like perfect square trinomials.
• Factor the expression accordingly.

For \(y^2 - 6yz + 9z^2\), recognizing it fits the form \(a^2 - 2ab + b^2\), we can factorize along these patterns and simplify it to \((y - 3z)^2\).
Verifying it last step is always a good practice, expand the simplified form to ensure that we obtain back the original polynomial.

Algebraic Patterns

Algebraic patterns are recurring structures in mathematics that help simplify complex equations. Some fundamental patterns include perfect square trinomials, difference of squares, and cubes.

The perfect square trinomial is a key algebraic pattern where:

• \(a^2 + 2ab + b^2 = (a + b)^2\)
• \(a^2 - 2ab + b^2 = (a - b)^2\)

Recognizing these patterns is crucial for efficient problem-solving. For example, with the expression \(y^2 - 6yz + 9z^2\), detecting it follows the perfect square trinomial pattern:\(a = y\) and \(b = 3z\). Thus, it simplifies to \((y-3z)^2\).

The more familiar you become with algebraic patterns, the easier it is to work through polynomial factorization and other algebraic manipulations.
Identifying and applying these patterns correctly provides powerful tools in math, leading to quicker and more accurate answers.