Problem 84 Factor. $$ 4 p^{2}+12 p q+9 ... [FREE SOLUTION]

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Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

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

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 $.

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 \[ 4p^2 + 12pq + 9q^2 = (2p)^2 + 2(2p)(3q) + (3q)^2 \].

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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91影视

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

Short Answer

Step by step solution

Identify the type of polynomial

Check for perfect square trinomial

Factor as a binomial square

Key Concepts

quadratic trinomial

perfect square trinomial

binomial square

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