Which of the following represents the factorization of the trinomial below?
[tex]x^2-6 x+9[/tex]
A. [tex](x+3)^2[/tex]
B. [tex](x+2)(x-3)[/tex]
C. [tex](x-2)(x-3)[/tex]
D. [tex](x-3)^2[/tex]
Answer (1)
Expand each of the given options.
Compare the expanded forms with the given trinomial.
Identify the option that matches the given trinomial.
The correct factorization is ( x − 3 ) 2 .
Explanation
Problem Analysis We are given the trinomial x 2 − 6 x + 9 and asked to find its factorization from the given options. We will expand each option to see which one matches the given trinomial.
Expanding the Options Let's expand each of the options:
Option A: ( x + 3 ) 2 = ( x + 3 ) ( x + 3 ) = x 2 + 3 x + 3 x + 9 = x 2 + 6 x + 9 . This does not match the given trinomial.
Option B: ( x + 2 ) ( x − 3 ) = x 2 − 3 x + 2 x − 6 = x 2 − x − 6 . This does not match the given trinomial.
Option C: ( x − 2 ) ( x − 3 ) = x 2 − 3 x − 2 x + 6 = x 2 − 5 x + 6 . This does not match the given trinomial.
Option D: ( x − 3 ) 2 = ( x − 3 ) ( x − 3 ) = x 2 − 3 x − 3 x + 9 = x 2 − 6 x + 9 . This matches the given trinomial.
Finding the Correct Factorization Comparing the expanded forms with the given trinomial x 2 − 6 x + 9 , we see that option D, ( x − 3 ) 2 , is the correct factorization.
Conclusion Therefore, the correct factorization of the trinomial x 2 − 6 x + 9 is ( x − 3 ) 2 .
Examples
Factoring quadratic trinomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and solve problems related to stress and strain. Imagine you are designing a rectangular garden with an area represented by the expression x 2 − 6 x + 9 . By factoring this expression to ( x − 3 ) 2 , you realize that the garden is a square with side length ( x − 3 ) . This helps you plan the layout and fencing required for the garden.
