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Chapter 1: Problem 30

Find the perfect square trinomial whose first two terms are given. $$x^{2}-12 x$$

Short Answer

Expert verified
The perfect square trinomial is \(x^2 - 12x + 36\).

Step by step solution

01

Identify the given terms

The given terms are the first two terms of the perfect square trinomial: \(x^2\) and \(-12x\).
02

Determine the coefficient of the linear term

The coefficient of the linear term \(-12x\) is \(-12\).
03

Divide the coefficient by 2

Divide \(-12\) by 2 to find the required constant for the middle term in the perfect square trinomial: \(-12 \div 2 = -6\).
04

Square the result

Square \(-6\) to get the constant term for the trinomial: \((-6)^2 = 36\).
05

Form the perfect square trinomial

Combine \(x^2\), \(-12x\), and the constant term \(36\) to form the perfect square trinomial: \(x^2 - 12x + 36\).

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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 three terms. These terms are separated by addition or subtraction operators.
An example of a trinomial is \(x^2 - 12x + 36\).
Here, there are three distinct parts: the squared term \(x^2\), the linear term \(-12x\), and the constant term \(36\).
Understanding each part is essential for working with trinomials.
  • Squared term: The term with the highest degree, containing a variable raised to the power of 2, like \(x^2\).
  • Linear term: The term with the variable raised to the power of 1, such as \(-12x\).
  • Constant term: A number without any variables, for example, \(36\).
square root
The square root of a number is a value that, when multiplied by itself, gives the original number.
For example, the square root of 36 is 6 because \(6 \times 6 = 36\).
In the process of forming a perfect square trinomial, the square root helps us find the constant term.
  • Step 1: Start with the linear coefficient. For example, in this problem, the coefficient of \(-12x\) is \(-12\).
  • Step 2: Divide this coefficient by 2. Here, \(-12 \, 梅 \, 2 = -6\).
  • Step 3: Square this result to get the constant term. In this case, \((-6)^2 = 36\).
factoring polynomials
Factoring polynomials involves writing a polynomial as a product of its factors.
For a perfect square trinomial like \(x^2 - 12x + 36\), it can be factored into \((x - 6)^2\).
This step is crucial in simplifying polynomials and solving polynomial equations.
To factor a perfect square trinomial, follow these steps:
  • Step 1: Identify the trinomial. For example, \(x^2 - 12x + 36\).
  • Step 2: Ensure it fits the form \(a^2 - 2ab + b^2\). Here, \(a = x\) and \(b = 6\), so the trinomial fits.
  • Step 3: Factor it as \((a - b)^2\), resulting in \((x - 6)^2\).
coefficients
Coefficients are numerical values that multiply the terms of a polynomial.
In the trinomial \(x^2 - 12x + 36\), we have different coefficients:
  • The coefficient of \(x^2\) is 1 (implied).
  • The coefficient of \(x\), the linear term, is \(-12\).
  • The constant term's coefficient is simply the constant value, \(36\).
Understanding coefficients helps in determining important features of the polynomial. Like, during formation of a perfect square trinomial:
  • Identify the linear term's coefficient.
  • Divide this by 2 to determine the necessary square root.
  • Square the result to find the perfect square trinomial's constant term.

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