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Chapter 6: Problem 84

Factor. $$ 81 p^{2}-36 p q+4 q^{2} $$

Short Answer

Expert verified
The expression factors to \((9p - 2q)^2\).

Step by step solution

01

Recognize the Structure

First, notice that the expression \(81p^2 - 36pq + 4q^2\) is a quadratic trinomial with respect to \(p\). It can be expressed in the standard form \(ax^2 + bxy + cy^2\), where \(a = 81\), \(b = -36\), and \(c = 4\).
02

Check if it's a Perfect Square Trinomial

To determine if it is a perfect square trinomial, we need to check if the first and last terms are squares and the middle term is twice the product of these square roots. Here, \(81p^2\) is \((9p)^2\) and \(4q^2\) is \((2q)^2\). The middle term \(-36pq\) should equal \(-2 \times 9p \times 2q = -36pq\). Hence, this is a perfect square trinomial.
03

Factor the Trinomial

Since the trinomial is a perfect square, we can express it as the square of a binomial. Therefore, \(81p^2 - 36pq + 4q^2\) can be factored as \((9p - 2q)^2\).
04

Verify the Factorization

To ensure the accuracy of our factorization, expand \((9p - 2q)^2\):\((9p - 2q)(9p - 2q) = 81p^2 - 18pq - 18pq + 4q^2 = 81p^2 - 36pq + 4q^2\). Since this matches the original expression, our factorization is correct.

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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 special form of a quadratic expression. It is the result of squaring a binomial. Understanding this concept is key to factoring expressions like the one in the exercise: \(81p^2 - 36pq + 4q^2\).

A trinomial is a perfect square if:
  • The first term is a square, such as \((9p)^2 = 81p^2\).
  • The last term is a square, like \((2q)^2 = 4q^2\).
  • The middle term is twice the product of the square roots of the first and last terms: \(-2 \times 9p \times 2q = -36pq\).
This arrangement ensures that the trinomial can be expressed as the square of a binomial. When you can identify this structure, you instantly know you can factor it into a binomial square.
binomial square
Understanding a binomial square is crucial when factoring perfect square trinomials. A binomial square takes the form \((a + b)^2\), which expands to \(a^2 + 2ab + b^2\). In our example, the binomial square that simplifies to the given trinomial is \((9p - 2q)^2\).

Factorization Process:
  • Identify the coefficients that form the square root of each perfect square in the trinomial: \(9p\) and \(2q\).
  • Verify that the middle term matches \(2ab\), which it does in this case as \(-2 \times 9p \times 2q = -36pq\).
  • The trinomial can thus be rewritten as \((9p - 2q)(9p - 2q)\), or simply \((9p - 2q)^2\).
Binomial squares simplify the task of factoring by associating complex polynomial expressions with simpler binomial terms.
quadratic expression
A quadratic expression is a polynomial with a degree of 2, commonly represented as \(ax^2 + bx + c\). Quadratics can usually be factored using various methods, but recognizing perfect square trinomials offers an efficient shortcut.

In the context of the given exercise, we deal with a quadratic trinomial given by \(81p^2 - 36pq + 4q^2\). Transforming this expression relies on observing its structure:
  • The terms align perfectly as a squared set of binomial terms - indicating it's a perfect square trinomial.
  • Quadratic expressions can often take different forms like \((x \pm y)^2\), perfect squares being one of the unique forms.
  • By identifying this structure, we鈥檙e able to factor it smartly with less guesswork involved, resulting in \((9p - 2q)^2\).
This highlights why knowing different forms of quadratic expressions and their characteristics enhances problem-solving flexibility.

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Most popular questions from this chapter

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 4 x^{2} y^{2}+4 x y^{2}+y^{2} $$

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The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ m^{2} n^{2}-9 m^{2}+3 n^{2}-27 $$

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ a^{2}(x-a)-b^{2}(x-a) $$

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ -9 x^{2}+6 x-1 $$
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