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Chapter 6: Problem 70

Factor completely. If a polynomial is prime, state this. $$ \frac{1}{4} a^{2}+\frac{1}{3} a b+\frac{1}{9} b^{2} $$

Short Answer

Expert verified
\( \frac{1}{36} (3a + 2b)^2 \)

Step by step solution

01

Identify a common factor

Examine the polynomial \(\frac{1}{4} a^{2} + \frac{1}{3} a b + \frac{1}{9} b^{2}\) for any common factors in the coefficients and variables.
02

Convert the polynomial coefficients to fractions with a common denominator

To make factoring easier, convert each term to have a common denominator. The least common multiple of the denominators 4, 3, and 9 is 36. Rewriting the polynomial we get: \(\frac{9}{36} a^{2} + \frac{12}{36} a b + \frac{4}{36} b^{2}\).
03

Simplify the polynomial

Now, factor out the common denominator \( \frac{1}{36} \) from the polynomial: \( \frac{1}{36} (9a^2 + 12ab + 4b^2) \).
04

Factor the quadratic expression inside the parentheses

Factor the quadratic expression \( 9a^2 + 12ab + 4b^2 \). Recognize it as a perfect square trinomial: \( (3a + 2b)^2 \). Therefore, \( 9a^2 + 12ab + 4b^2 = (3a + 2b)^2 \).
05

Write the final factored form

Combine the factored expression to the common denominator: \( \frac{1}{36} (3a + 2b)^2 \).

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Key Concepts

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

Quadratic Expression
A quadratic expression is a polynomial of degree 2, which means the highest power of the variable is squared. For instance, in the polynomial \(\frac{1}{4} a^{2} + \frac{1}{3} a b + \frac{1}{9} b^{2}\), the term \(a^2\) signifies that it's a quadratic expression. Understanding the structure of these expressions helps in factoring them. Often it appears in the form \(ax^2 + bxy + cy^2\), where \(a, b,\) and \(c\) are coefficients.
Common Denominator
When dealing with fractions, a common denominator is necessary to make calculations more straightforward. It's especially useful in polynomial expressions like \(\frac{1}{4} a^{2} + \frac{1}{3} a b + \frac{1}{9} b^{2}\). To add, subtract, or compare fractions, their denominators must be the same. For example, the least common multiple of 4, 3, and 9 is 36, making it the common denominator in this case. This common base simplifies subsequent steps in factoring or simplifying expressions.
Perfect Square Trinomial
A perfect square trinomial takes the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). It expands from either \((a + b)^2\) or \((a - b)^2\). For our polynomial, \(9a^2 + 12ab + 4b^2\) fits this description. We can recognize this perfect square by checking the middle term (12ab) to see if it equals 2 times the product of the square roots of the first and last terms (3a and 2b). This recognition allows us to rewrite \(9a^2 + 12ab + 4b^2\) as \((3a + 2b)^2\).
Polynomial Factorization
Polynomial factorization is the process of expressing a polynomial as a product of its factors. Applying this to \(\frac{1}{4} a^{2} + \frac{1}{3} a b + \frac{1}{9} b^{2}\), we first handle the denominators to standardize them, and then identify patterns like perfect square trinomials. The original polynomial gets simplified to \(\frac{1}{36} (9a^2 + 12ab + 4b^2)\) and eventually factored into \(\frac{1}{36} (3a + 2b)^2\). This process makes the polynomial easier to handle, both in theoretical and practical applications.
Common Factor
Identifying a common factor in a polynomial involves finding a term that is present in all parts of the expression. In \(\frac{1}{4} a^{2} + \frac{1}{3} a b + \frac{1}{9} b^{2}\), after adjusting for a common denominator, we simplify to \(\frac{1}{36} (9a^2 + 12ab + 4b^2)\). Here, \(\frac{1}{36}\) is a common factor that helps to streamline the process. Recognizing and factoring out common elements simplifies complex expressions, making it easier to see underlying patterns and complete the factorization process effectively.

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