What is the factored form of the polynomial [tex]x^2-12 x+27[/tex]?

A. [tex](x+4)(x+3)[/tex]
B. [tex](x-4)(x+3)[/tex]
C. [tex](x+9)(x+3)[/tex]
D. [tex](x-9)(x-3)[/tex]

The factored form of the polynomial x 2 − 12 x + 27 is ( x − 9 ) ( x − 3 ) . This is because -9 and -3 are the numbers that multiply to 27 and add to -12. Thus, the correct choice is option D.
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Answered by Anonymous | 2025-07-04

Find two numbers that multiply to 27 and add up to -12.
The factor pairs of 27 are (1, 27) and (3, 9). Consider negative factor pairs: (-1, -27) and (-3, -9).
Check which pair adds up to -12: -3 + (-9) = -12.
The factored form is ( x − 9 ) ( x − 3 ) ​ .

Explanation

Understanding the Problem We are given the quadratic polynomial x 2 − 12 x + 27 and asked to find its factored form. Factoring a quadratic means expressing it as a product of two binomials, like ( x + a ) ( x + b ) , where a and b are constants. When we expand ( x + a ) ( x + b ) , we get x 2 + ( a + b ) x + ab . So, we need to find two numbers a and b such that a + b = − 12 and ab = 27 .

Finding the Factors We need to find two numbers that multiply to 27 and add up to -12. Let's list the factor pairs of 27: (1, 27) and (3, 9). Since the middle term is -12, we need to consider negative factor pairs: (-1, -27) and (-3, -9).

Checking the Sum Now, let's check which of these pairs adds up to -12:



-1 + (-27) = -28
-3 + (-9) = -12 So, the correct pair is -3 and -9.


Writing the Factored Form Therefore, the factored form of the polynomial is ( x − 3 ) ( x − 9 ) .

Final Answer The factored form of the polynomial x 2 − 12 x + 27 is ( x − 9 ) ( x − 3 ) .


Examples
Factoring quadratic equations is a fundamental skill in algebra with numerous real-world applications. For instance, consider a rectangular garden whose area is represented by the quadratic expression x 2 − 12 x + 27 . By factoring this expression into ( x − 9 ) ( x − 3 ) , we determine that the dimensions of the garden are ( x − 9 ) and ( x − 3 ) . This allows us to calculate the amount of fencing needed (perimeter) or the amount of soil required (area) for different values of x . Factoring helps in optimizing the garden's design based on available resources and space.

Answered by GinnyAnswer | 2025-07-04