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Chapter 1: Problem 105

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

Short Answer

Expert verified
The expression \( t^2 - 6t + 9 \) factors to \((t - 3)^2\).

Step by step solution

01

Identify the Quadratic Form

The expression given is a quadratic trinomial: \( t^2 - 6t + 9 \). Here, the coefficients are: \(a = 1\), \(b = -6\), and \(c = 9\).
02

Check for Perfect Square Trinomial

A perfect square trinomial takes the form \((t + d)^2 = t^2 + 2dt + d^2\). Compare with \( t^2 - 6t + 9 \). Notice that \((-3)^2 = 9\) and \(2(-3) = -6\), which matches our trinomial.
03

Write the Expression as a Square of a Binomial

Since \( t^2 - 6t + 9 \) is a perfect square trinomial, we can write it as \((t - 3)^2\).

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Key Concepts

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

Quadratic Expressions
A quadratic expression is a polynomial with a degree of two. This means its highest exponent is two. It usually takes the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). Quadratic expressions are important in algebra as they represent parabolas when graphed on a Cartesian plane. These expressions can describe a wide range of real-world phenomena, such as projectile motion or optimization problems in economics.
  • In our example of \( t^2 - 6t + 9 \), \( a = 1 \), \( b = -6 \), and \( c = 9 \).
  • The standard form can help identify key properties of the quadratic, such as its vertex, axis of symmetry, and direction of opening (upwards or downwards).
Recognizing the structure of a quadratic expression is the first step towards factoring, solving, or graphing them. Understanding these basics lays the groundwork for more advanced algebraic techniques.
Perfect Square Trinomial
A perfect square trinomial is a special type of quadratic expression. It can be written as the square of a binomial. This means it takes the form \((t + d)^2\), which expands to \(t^2 + 2dt + d^2\). Identifying a perfect square trinomial can simplify factoring significantly.
  • In the given expression \( t^2 - 6t + 9 \), we can see it fits the pattern of a perfect square trinomial.
  • Here, \( d = -3 \) because \((-3)^2 = 9\) and \(2(-3) = -6\), making \(2dt\) match our middle term.
The advantage of identifying these trinomials is they can be quickly factored into \((t + d)^2\), making them simpler to work with in equations. This recognition comes from practice with identifying patterns, which is a critical skill in algebra.
Binomial Factorization
Binomial factorization involves rewriting a quadratic expression as the product of two binomial expressions. Once we identify a perfect square trinomial, the factorization becomes relatively straightforward. In the example \( t^2 - 6t + 9 \), recognizing it as a perfect square trinomial allows us to write it as \((t - 3)^2\).
  • This factorization is valid because squaring the binomial \((t - 3)\) will expand back to the original trinomial.
  • For students, this means checking the structure and recognizing patterns can vastly simplify the process.
Binomial factorization is a powerful tool because it not only makes solving quadratics easier but also simplifies integration, differentiation, and simplifying algebraic fractions in calculus and advanced algebra.

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Most popular questions from this chapter

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