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Chapter 1: Problem 33

Find the perfect square trinomial whose first two terms are given. $$w^{2}+\frac{1}{2} w$$

Short Answer

Expert verified
The perfect square trinomial is \(w^2 + \frac{1}{2} w + \frac{1}{16}\).

Step by step solution

01

Identify the first and second terms

The given terms are: First term: \(w^2\)Second term: \(\frac{1}{2} w\)
02

Determine the coefficient of the second term

The coefficient of the second term is \( \frac{1}{2} \).
03

Divide the coefficient of the second term by 2

Divide \( \frac{1}{2} \) by 2 to find the middle term coefficient: \( \frac{1}{2} \div 2 = \frac{1}{4} \).
04

Square the result from Step 3

Square \( \frac{1}{4} \) to find the constant term of the perfect square trinomial: \[\left( \frac{1}{4} \right) ^2 = \frac{1}{16}.\]
05

Write the perfect square trinomial

Combine the original terms and the constant term found in Step 4: \[w^2 + \frac{1}{2} w + \frac{1}{16}.\]
06

Verify the perfect square trinomial

Check that the trinomial can be written as the square of a binomial: \[\left( w + \frac{1}{4} \right) ^2 = w^2 + \frac{1}{2} w + \frac{1}{16}.\]

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Key Concepts

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

Trinomial
A trinomial is a type of polynomial with exactly three terms. In general, these terms are a combination of variables and constants combined using addition or subtraction. For instance, the expression you worked with,
  • \[w^2 + \frac{1}{2}w + \frac{1}{16}\]
is a trinomial because it contains three distinct terms. Understanding trinomials is crucial for solving algebraic equations and simplifying expressions. Let's quickly break down each term:
  • \(w^2\): This is the quadratic term.
  • \(\frac{1}{2}w\): This is the linear term.
  • \(\frac{1}{16}\): This is the constant term.
Each of these terms plays a role in forming the perfect square trinomial, which is essential for various algebraic processes, including factoring.
Coefficients
Coefficients are the numerical factors in terms of an algebraic expression. For example, in the term
  • \( \frac{1}{2}w \), the coefficient is \( \frac{1}{2} \).
They indicate how many times the variable (in this case, \( w \)) is multiplied. Identifying coefficients is a fundamental step in algebra, particularly when forming trinomials. Let's see how to work with them:
  • If you have a term like \( \frac{1}{2}w \), divide the coefficient (\( \frac{1}{2} \)) by 2 to assist in completing the square. For example, \( \frac{1}{2} \div 2 = \frac{1}{4} \).
  • Next, square this result to find the constant term: \[ \left( \frac{1}{4} \right)^2 = \frac{1}{16}. \]
This attention to coefficients ensures accuracy in forming your final trinomial.
Factoring
Factoring is the process of breaking down an expression into simpler terms (factors) that, when multiplied together, produce the original expression. For the perfect square trinomial
  • \[ w^2 + \frac{1}{2}w + \frac{1}{16} \]
our goal is to express it as the square of a binomial. This involves recognizing patterns and using algebraic techniques to simplify. Following the exercise:
  • Write the given trinomial as a product of two binomials: Find two binomials that multiply to form the trinomial.
  • Verify your factorization: Check if squaring the binomial gives back the original trinomial.
For example, we can rewrite
  • \[ w^2 + \frac{1}{2}w + \frac{1}{16}\]
as
  • \[ \left( w + \frac{1}{4} \right)^2. \]
When expanded, it equals the original trinomial, confirming that the factorization is correct. Mastering factoring helps solve equations more effectively and simplifies complex algebraic expressions.

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