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Chapter 5: Problem 84

Factor. $$ 4 p^{2}+12 p q+9 q^{2} $$

Short Answer

Expert verified
The factored form is \( (2p + 3q)^2 \).

Step by step solution

01

Identify the type of polynomial

Examine the given polynomial to determine its type. The polynomial is a quadratic trinomial of the form \( ax^2 + bx + c \). Here, \( a = 4 \), \( b = 12 \), and \( c = 9 \).
02

Check for perfect square trinomial

A perfect square trinomial is of the form \[ a^2 + 2ab + b^2 \]. For the given polynomial, verify if it can be written in this form. Notice that: \( 4p^2 = (2p)^2 \), \( 12pq = 2(2p)(3q)\), and \( 9q^2 = (3q)^2 \). Hence, the polynomial \[ 4p^2 + 12pq + 9q^2 = (2p)^2 + 2(2p)(3q) + (3q)^2 \].
03

Factor as a binomial square

Since the polynomial \[ 4p^2 + 12pq + 9q^2 \] fits the form of \[ (a + b)^2 \], it can be factored as \[ (2p + 3q)^2 \].

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Key Concepts

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

quadratic trinomial
Understanding a quadratic trinomial is critical in polynomial factoring. It is an algebraic expression of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our given example, \[4p^2 + 12pq + 9q^2\], we identify that \(a = 4\), \(b = 12\), and \(c = 9\). The highest exponent in such an expression is 2, making it quadratic.

Here are a few tips to identify a quadratic trinomial:
  • There are three terms in the polynomial.
  • The highest degree (power of the variable) is 2.
  • The middle term is linear (degree 1), and the last term is a constant.
Identifying these features is the first step in the factoring process.
perfect square trinomial
A perfect square trinomial is a special type of quadratic trinomial. Its form is \[a^2 + 2ab + b^2\]. Recognizing this form allows us to factor the trinomial into a binomial square easily.

Let us break down our polynomial again to see why it fits the form:
  • \[4p^2 \ = (2p)^2\] - This is the square of \(2p\).
  • \[12pq = 2 \cdot 2p \cdot 3q\] - This is twice the product of \(2p\) and \(3q\).
  • \[9q^2 = (3q)^2\] - This is the square of \(3q\).
Thus, \[4p^2 + 12pq + 9q^2 \ = (2p)^2 + 2(2p)(3q) + (3q)^2\], showing that our original polynomial is indeed a perfect square trinomial.
binomial square
Once you identify that a polynomial is a perfect square trinomial, you can factor it into a binomial square. A binomial square is an expression of the form \[ (a + b)^2 \].

For our polynomial \[4p^2 + 12pq + 9q^2\], since we have shown that it can be written as \[ (2p)^2 + 2(2p)(3q) + (3q)^2\], it can be factored into \[ (2p + 3q)^2 \]. This is our binomial square.

Here's a quick recap to form a binomial square:
  • Ensure the original polynomial is a perfect square trinomial.
  • Identify \(a\) and \(b\).
  • Write the binomial as \[ (a + b)^2 \].
So, for our example, from \[ 4p^2 + 12pq + 9q^2\], we derive the binomial square \[ (2p + 3q)^2 \].

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