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Chapter 1: Problem 107

Factor the expression completely. $$t^{2}-6 t+9$$

Short Answer

Expert verified
The expression factors as \((t - 3)^2\).

Step by step solution

01

Identify the structure of the expression

The expression is a quadratic, written in the form \( t^2 - 6t + 9 \). This can potentially be factored using the perfect square trinomial method since the first and last terms are perfect squares.
02

Express the trinomial in perfect square form

The expression \( t^2 - 6t + 9 \) can often be factored as a perfect square trinomial. Let's verify by checking if it can take the form \( (t - a)^2 \). Expanding the perfect square, we have: \ (t - 3)^2 = t^2 - 2 \times 3 \times t + 3^2 = t^2 - 6t + 9. \ This matches our original expression.
03

Write the expression as a squared binomial

Since the expression can indeed be represented as \( (t - 3)^2 \), this indicates that it factors into a squared binomial.

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Key Concepts

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

Perfect Square Trinomial
Understanding a perfect square trinomial can make factoring certain quadratic expressions much simpler. A perfect square trinomial arises when a binomial is multiplied by itself. The general form is \((a \, \pm \, b)^2 = a^2 \, \pm \, 2ab \, + \, b^2\).
When you recognize a quadratic expression, such as \(t^2 - 6t + 9\), you can see if it's a perfect square trinomial by checking if it fits this pattern.
  • The first term \(t^2\) is \(a^2\), where \(a = t\).
  • The last term \(9\) is \(b^2\), where \(b = 3\).
  • The middle term \(-6t\) is \(-2ab = -2 \times t \times 3\), confirming the pattern.
Recognizing this pattern allows you to factor \(t^2 - 6t + 9\) as \((t - 3)^2\), indicating it's a perfect square trinomial.
Quadratic Expressions
Quadratic expressions are polynomials that contain terms up to the second degree. Typically, they are written in the standard form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our example, \(t^2 - 6t + 9\), this translates to:
  • \(a = 1\), meaning the leading coefficient of \(t^2\) is 1.
  • \(b = -6\), the coefficient of \(t\).
  • \(c = 9\), the constant term.
To factor this expression, the perfect square trinomial technique is effective because of special relationships among these coefficients. Recognizing the connection between the terms can simplify solving quadratic expressions in algebra.
Binomial Squares
A binomial square results from squaring a single binomial, which can be two numbers or expressions added or subtracted. The expression \((a - b)^2\) is a common form, expanding to \(a^2 - 2ab + b^2\). In the exercise, the goal was to recognize that \(t^2 - 6t + 9\) was a perfect square of a binomial.
  • Identifying binomial squares helps in quickly factoring expressions.
  • In our case, \((t - 3)^2\) is the required binomial resulting from the initial quadratic.
  • This allows us to break down solving into simpler arithmetic and algebraic steps.
Understanding binomial squares is fundamental in algebra, as it offers an efficient method to handle expressions that might otherwise seem complicated.

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Most popular questions from this chapter

Plot the points \(M(6,8)\) and \(A(2,3)\) on a coordinate plane. If \(M\) is the midpoint of the line segment \(A B,\) find the coordinates of \(B\) Write a brief description of the steps you took to find \(B\) and your reasons for taking them.

Test the equation for symmetry. $$y=x^{2}+|x|$$

Solve the equation for the variable \(x\). The constants \(a\) and \(b\) represent positive real numbers. $$\sqrt{x+a}+\sqrt{x-a}=\sqrt{2} \sqrt{x+6}$$

Test the equation for symmetry. $$y=x^{3}+10 x$$

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. $$\frac{1.295643 \times 10^{9}}{\left(3.610 \times 10^{-17}\right)\left(2.511 \times 10^{6}\right)}$$
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