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Chapter 5: Problem 100

Factor. If the polynomial is prime, so indicate. $$9 a^{2}-6 a c-c^{2}$$

Short Answer

Expert verified
The factorised form of \(9a^{2}-6ac-c^{2}\) is \((3a - c)^{2}\).

Step by step solution

01

Recognize structure

First, identify the structure of the equation. It appears that \(9a^{2}-6ac-c^{2}\) is a perfect square trinomial in the form of \(a^{2}-2ab+b^{2}\).
02

Identify a, b, and c

Next, identify a, b, and c from this trinomial. From the structure, it's apparent that \(a = 3a\), \(b = c\) and \(2ab = 6ac\).
03

Factor the trinomial

Knowing that the form is \(a^{2}-2ab+b^{2} = (a - b)^{2}\), you can substitute a and b into the second part of the formula to find that \(9a^{2}-6ac-c^{2} = (3a-c)^{2}\).

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Key Concepts

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

Perfect Square Trinomial
A perfect square trinomial is a specific kind of polynomial that can be rewritten as a squared binomial. For example, a generic form of such a trinomial is \( a^2 - 2ab + b^2 \). Notice how the last term \( b^2 \) and the first term \( a^2 \) make it a perfect square. The middle term \( -2ab \) is twice the product of the square roots of the first and last term.
When factoring a trinomial like \( 9a^2 - 6ac - c^2 \), you recognize if it's a perfect square by checking these conditions. If they match, then the expression can be rewritten as \( (3a - c)^2 \). The beauty of this form lies in how it showcases symmetry and structure, providing simplicity in factoring.
Polynomial Structure
The structure of a polynomial is crucial in determining how to manipulate or factor it. A polynomial like \( 9a^2 - 6ac - c^2 \) is generally expressed in the form \( ax^2 + bxy + cy^2 \). Each term plays a significant role in categorizing the type and structure of the polynomial.
  • First term \( 9a^2 \) comes from squaring \( 3a \).
  • Middle term \( -6ac \) indicates the interaction between the variables \( a \) and \( c \).
  • Last term \( -c^2 \) is the square of \( c \).
Knowing these allows you to establish patterns or parallels with standard forms like the perfect square trinomial.
Identifying Coefficients
Identifying coefficients in a polynomial is the first step toward factoring or simplifying expressions. These coefficients, usually numbers before variables, tell important details about the polynomial's structure. In the expression \( 9a^2 - 6ac - c^2 \), the coefficients are:
  • 9 for \( 9a^2 \)
  • -6 for \( -6ac \)
  • -1 for \( -c^2 \)
Understanding these coefficients helps in identifying relationships between terms. In this example, the specific coefficients led to recognizing it as a perfect square trinomial. By breaking down and analyzing coefficients, factoring complex expressions becomes more straightforward and manageable.

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Most popular questions from this chapter

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