Problem 3 - Elementary Algebra

Question

Use a pattern to factor. Check. Identify any prime polynomials. $$ y^{2}-12 y+36 $$

Step-by-Step Solution

Answer

The polynomial \( y^{2} - 12y + 36 \) factors to \( (y - 6)^{2} \) and is not a prime polynomial.

Step 1 - Identify the polynomial form

Recognize that the given polynomial is a quadratic polynomial in the form of \( y^{2} - 12y + 36 \)

Step 2 - Look for a perfect square

Notice that both the first term \( y^{2} \) and the last term \( 36 \) are perfect squares. Rewrite the polynomial as:\( (y)^{2} - 2 \times 6 \times y + (6)^{2} \)

Step 3 - Factor the perfect square trinomial

Use the pattern of perfect square trinomials \( a^{2} - 2ab + b^{2} = (a - b)^{2} \). Therefore, \( y^{2} - 12y + 36 = (y - 6)^{2} \)

Step 4 - Identify prime polynomials

Check if the factored form \( (y - 6)^{2} \) is further factorable over the integers. It isn't, so the polynomial is factored completely.

Key Concepts

Perfect Square Trinomials

In factoring quadratic polynomials, perfect square trinomials play a vital role. Perfect square trinomials follow the form:
. To recognize these, notice that the first term and the last term are squares of some expressions. For instance,
\((y^{2} - 12y + 36)\) is a perfect square trinomial.
In this polynomial:

• The first term \(y^{2}\) is
  the square of \(y\)
;
• The last term \(36 \ (6^{2})\) is the square of \(6\)
;
• The middle term: \(-12y\) is twice the product of \(y\) and \(6\), written as 2 \(ab\).

Thus, our equation can be written as: \((y - 6)^2\). Understanding perfect square trinomials simplifies the factoring process and helps in recognizing instantly factorable expressions.

Prime Polynomials

Prime polynomials are analogous to prime numbers. They cannot be factored further over the integers. When working with quadratic polynomials, after attempting all possible factoring methods, if no factorization is possible, the polynomial is considered prime.
In the given polynomial \(y^{2} - 12y + 36\):

• We checked if it could be factored further after recognizing it as a perfect square trinomial.
• Upon breaking it down, we saw that \((y - 6)^{2} \) was its simplest form.
This means that it cannot be factored further. Hence, the problem doesn't contain a prime polynomial because \( (y - 6)^2 \) is factorable.

Knowing when a polynomial is prime saves a lot of effort and time in the mathematics factoring process.

Factoring Patterns

Factoring patterns are templates or