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Chapter 6: Problem 14

Determine whether each of the following is a perfectsquare trinomial. $$ x^{2}+4 x+4 $$

Short Answer

Expert verified
Yes, it is a perfect square trinomial.

Step by step solution

01

Identify the form of the trinomial

Check if the trinomial is in the form of a^2 + 2ab + b^2.Here: a = xa^2 = x^2
02

Check the middle term

Identify 2ab and check if it equals the middle term of the given expression. Given middle term: 4xAssuming b = 2: 2ab = 2 * x * 2 = 4x
03

Check the constant term

Identify b^2 and check if it equals the constant term of the given expression. Given constant term = 4Assuming b = 2: b^2 = 2^2 = 4
04

Conclusion

Since the given expression matches the form a^2 + 2ab + b^2, it is a perfect square trinomial.

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Key Concepts

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

headline of the respective core concept
A trinomial is a type of polynomial that consists of three terms, which are usually separated by addition or subtraction. In general, a polynomial is an algebraic expression made up of variables, coefficients, and exponents, connected by addition, subtraction, or multiplication.
When dealing with trinomials, it's essential to understand their structure. A typical trinomial looks like this: ax^2 + bx + c.
The terms are:
  • ax^2: This term is called the quadratic term because it involves the variable raised to the power of 2.
  • bx: This term is known as the linear term, involving the variable raised to the power of 1.
  • c: This is the constant term. It does not have a variable attached to it.
Each part of a trinomial contributes to its overall structure and the methods you can use to factor or simplify it.
headline of the respective core concept
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols.
One of the main goals in algebra is to solve equations and simplify expressions. Algebra involves working with variables (symbols that represent unknown values) and constants (known values).
The primary operations include addition, subtraction, multiplication, and division.
In the context of the given trinomial, some key algebraic steps include:
  • Identifying coefficients and constants: From the trinomial x^2 + 4x + 4, identify a = 1, b = 4, and c = 4.
  • Recognizing patterns: We see that this trinomial can be expressed in the form a^2 + 2ab + b^2.
Understanding these fundamental algebraic concepts helps you follow the steps needed to determine whether a trinomial is a perfect square.
headline of the respective core concept
Factoring polynomials is a critical skill in algebra that involves breaking down a polynomial into simpler parts, or factors, that, when multiplied together, give the original polynomial.
For trinomials, factoring is often about recognizing specific patterns or forms.
For instance, in the given exercise, the trinomial x^2 + 4x + 4 follows the form of a perfect square trinomial.
Perfect square trinomials have the format: a^2 + 2ab + b^2, and factor into the form: (a + b)^2.
Here鈥檚 how you can factor the given trinomial:
  • Identify a and b from x^2 + 4x + 4: In this case, a = x and b = 2.
  • Rewrite the trinomial using these values to confirm the structure: x^2 + 2 * x * 2 + 2^2.
  • Factor it into (x + 2)^2.
Crucially, recognizing and factoring perfect square trinomials simplify many algebraic processes, such as solving equations and integrating expressions.

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