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Chapter 6: Problem 46

Factor each trinomial completely. $$ 4 z^{2}-12 z w+9 w^{2} $$

Short Answer

Expert verified
(2z - 3w)^2

Step by step solution

01

Identify the trinomial structure

The given expression is in the form of a quadratic trinomial: 4z^{2} - 12zw + 9w^{2}
02

Recognize perfect square trinomials

Observe that the given trinomial can be written in the form (a - b)^2 if a^2 is the first term, b^2 is the last term, and 2ab is the middle term.
03

Rewrite the trinomial

Notice that: 4z^{2} = (2z)^{2}, 9w^{2} = (3w)^{2}, and -12zw = 2(2z)(-3w) . Thus it can be rewritten as: (2z - 3w)^2 .
04

Factor completely

It follows from Step 3 that: (2z - 3w)^2 .

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Key Concepts

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

quadratic trinomial
A quadractic trinomial is a polynomial with three terms. The general form looks like this:
ax^2 + bx + c
This means it has one quadratic term (x^2degree), one linear term (xdegree), and one constant term. In our example, the quadratic trinomial is 4z^2 - 12zw + 9w^2.
Quadratic trinomials are most often seen in algebra, and learning to factor them is very useful. Factoring means breaking down the expression into simpler expressions that can be multiplied together to get the original expression.
perfect square trinomial
A perfect square trinomial is a special type of quadratic trinomial. It鈥檚 formed when a binomial is squared. The general form is:
(a - b)^2 = a^2 - 2ab + b^2
For example, in our case: 4z^2 - 12zw + 9w^2. We noticed that: 4z^2 is (2z)^2, 9w^2 is (3w)^2, and -12zw is 2(2z)(-3w).
When you match 4z^2 = (2z)^2, 9w^2 = (3w)^2, and -12zw = 2(2z)(-3w),you can rewrite the whole expression as (2z - 3w)^2.This makes it easy to factor the trinomial completely.
factoring trinomials
Factoring trinomials is all about rewriting the expression into its simplest form. This can help solve equations or simplify expressions. Let鈥檚 understand the steps to factor trinomials:
  • First, identify if you have a quadratic trinomial.
  • Next, check if the trinomial is a perfect square. This means identifying terms like a^2, b^2,and 2ab
  • Finally, rewrite the trinomial using the factor form: (a - b)^2.
The example we worked on, 4z^2 - 12zw + 9w^2, followed these steps:
  • Recognize the structure: Quadratic trinomial 4z^2 - 12zw + 9w^2
  • Identify it as a perfect square: (2z - 3w)^2
  • Rewrite and factor: (2z - 3w)^2
Understanding these concepts makes factoring trinomials much easier and useful in various algebra problems.

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Most popular questions from this chapter

Factor completely. If the polynomial cannot be factored, write prime. $$ -32+14 x+x^{2} $$

An expression is factored when it is written as a product, not a sum. Which of the following are not factored? \(2 k^{2}(5 k)\)

Factor by grouping. \(6-3 x-2 y+x y\)

Write in factored form by factoring out the greatest common factor. \(16 z^{4}+24 z^{2}\)

The middle term of each trinomial has been rewritten. Now factor by grouping. $$ \begin{aligned} 6 x^{2}+13 x+6 \\ =6 x^{2}+9 x+4 x+6 \end{aligned} $$
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