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Chapter 6: Problem 11

For Problems \(1-12\), factor each of the perfect square trinomials. (Objective 1 ) $$ 16 x^{2}-24 x y+9 y^{2} $$

Short Answer

Expert verified
The expression factors to \((4x - 3y)^2\).

Step by step solution

01

Identify the Trinomial Structure

The given expression is structured as a trinomial: \(16x^2 - 24xy + 9y^2\). Notice that the first term, \(16x^2\), and the last term, \(9y^2\), are perfect squares.
02

Recognize the Perfect Squares

Identify the square roots of the first and last terms: \(\sqrt{16x^2} = 4x\) and \(\sqrt{9y^2} = 3y\). This suggests the binomials \((4x ? 3y)\) might be the factors.
03

Check the Middle Term

The middle term, \(-24xy\), should equal \(-2\times (4x)\times (3y)\). Calculate: \(-2 \times 4x \times 3y = -24xy\), which matches the middle term of the original trinomial.
04

Write the Factored Form

Since the middle term checks out, the trinomial \(16x^2 - 24xy + 9y^2\) factors into \((4x - 3y)^2\).
05

Final Step: Verify the Solution

Expand \((4x - 3y)^2\) to ensure it equals the original expression: \( (4x - 3y)(4x - 3y) = 16x^2 - 12xy - 12xy + 9y^2 = 16x^2 - 24xy + 9y^2\). The factorization is correct.

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Key Concepts

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

Understanding Trinomial Structure
A trinomial is a type of polynomial that consists of three terms. In our exercise, the trinomial is given as \(16x^2 - 24xy + 9y^2\). Trinomials usually present themselves in the form \(ax^2 + bx + c\). Here, upon closer inspection, we can see that:
  • \(16x^2\) is our first term, representing \(a\), which in this case equals \(a^2\).
  • \(-24xy\) is the second term, sometimes known as the cross term or middle term which acts as \(b\).
  • \(9y^2\) is the third term, acting as \(c\), represented as \(b^2\).
Both the first and last terms are perfect squares, which provides a hint that this trinomial could be factored. Recognizing this structure is key in factoring and impacts the methodology we choose to solve it.
Identifying Perfect Squares
In order to factor perfect square trinomials, we must first identify the perfect squares. In the given trinomial \(16x^2 - 24xy + 9y^2\):
  • Notice that \(16x^2\) is a perfect square with \(\sqrt{16x^2} = 4x\).
  • Similarly, \(9y^2\) is a perfect square with \(\sqrt{9y^2} = 3y\).
When we find these roots, they pave the way for us to break down the trinomial into binomial factors. These roots point to our next step in testing if our expression is truly a perfect square trinomial. They help propose what the factors could potentially look like: namely, \((4x ? 3y)^2\). Understanding how to recognize these perfect squares makes the factoring process smoother and more intuitive.
Deriving the Factored Form
The final objective in factoring perfect square trinomials is to derive the expression in its factored form. With the perfect squares we have identified, we propose: \((4x - 3y)^2\).Now, let's ensure it reflects the original problem:
  • We expand \((4x - 3y)^2\) to yield \((4x - 3y)(4x - 3y)\).
  • This outputs: \(16x^2 - 12xy - 12xy + 9y^2\).
  • When simplified, it becomes \(16x^2 - 24xy + 9y^2\).
This confirms our factored form correctly represents the original trinomial. Overall, understanding how to transition from recognizing trinomial structure to identifying perfect squares and finally arriving at the factored form requires practice. The more you engage with these steps, the easier it becomes to factor trinomials efficiently.

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Most popular questions from this chapter

\(-x^{2}-2 x+24=0\)

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ 30 n^{2}-n-1=0 $$

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ x^{3}=-4 x^{2} $$

What does the expression "not factorable using integers" mean to you?

For Problems \(71-88\), set up an equation and solve each problem. (Objective 4) Find two numbers whose product is \(-1\). One of the numbers is three more than twice the other number.
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