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Chapter 5: Problem 24

Factor each polynomial. $$ 9 y^{2}+6 y z+z^{2} $$

Short Answer

Expert verified
The polynomial factors to (3y + z)^2.

Step by step solution

01

Identify the Polynomial

Given polynomial is 9y^2 + 6yz + z^2. Notice that this polynomial is a quadratic trinomial in terms of y.
02

Recognize Perfect Square Trinomial

Check if the polynomial can be written in the form of (a + b)^2 = a^2 + 2ab + b^2.
03

Find the Terms a and b

Compare 9y^2 + 6yz + z^2 with a^2 + 2ab + b^2. Identify the terms: a^2 = (3y)^2, 2ab = 6yz, and b^2 = z^2.
04

Verify the Middle Term

Make sure that the middle term fits: 2ab = 6yz, so 2(3y)(z) = 6yz, which matches the middle term.
05

Factor the Polynomial

Since it matches the pattern of a perfect square trinomial, factor it as (3y + z)^2.
06

Write the Final Answer

So, the factored form of 9y^2 + 6yz + z^2 is (3y + z)^2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic trinomial
A quadratic trinomial is a type of polynomial that involves three terms, with the highest degree being 2. Specifically, it can be written in the general form:
  • ax^2 + bx + c
Here, 'a', 'b', and 'c' are coefficients, with 'x' being the variable. In the given example, 9y^2 + 6yz + z^2, the variable 'y' is squared making it a quadratic polynomial. Importantly, it has three terms thus forming a trinomial. Breaking it down:
  • 9y^2 is the quadratic term
  • 6yz is the linear term
  • z^2 is the constant term
Each term holds a role in the structure necessary for factoring the polynomial efficiently.
perfect square trinomial
A perfect square trinomial is a special case of quadratic trinomials. It takes the specific form:
  • (a + b)^2 = a^2 + 2ab + b^2
Recognizing this pattern is key to factoring it easily. The example polynomial, 9y^2 + 6yz + z^2, matches this structure. Let's compare:
  • In 9y^2 + 6yz + z^2: a^2 = (3y)^2, 2ab = 6yz, and b^2 = z^2
  • Therefore, a = 3y, and b = z
Verifying the middle term is crucial, and here, 2ab = 2(3y)(z) = 6yz, confirming the match. By identifying this structure, you can quickly factor the polynomial as: (3y + z)^2
factoring techniques
Factoring techniques simplify solving quadratic trinomials by breaking them down into simpler expressions. Recognizing patterns like perfect square trinomials makes this process easier. For our example, since 9y^2 + 6yz + z^2 can be identified as a perfect square trinomial, follow these steps:
  • Compare the polynomial to a^2 + 2ab + b^2
  • Identify 'a' and 'b' from the polynomial
  • Verify the middle term 2ab, ensure it matches the polynomial's middle term
Once confirmed, rewrite the polynomial in factored form: (3y + z)^2. Mastery of these techniques not only makes solving polynomial equations straightforward, but also builds a solid foundation for more advanced algebra topics. Practice various factoring methods often to enhance your skills.

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