Which of the following represents the factorization of the trinomial below?
[tex]$x^2+9 x-22$[/tex]
A. [tex]$(x+11)(x-2)$[/tex]
B. [tex]$(x+20)(x-2)$[/tex]
C. [tex]$(x-20)(x-2)$[/tex]
D. [tex]$(x-11)(x-2)$[/tex]
Answer (2)
Find two numbers that multiply to -22 and add up to 9.
The numbers are -2 and 11.
Write the factorization as ( x − 2 ) ( x + 11 ) or ( x + 11 ) ( x − 2 ) .
The correct factorization is ( x + 11 ) ( x − 2 ) .
Explanation
Understanding the Problem We are given the quadratic trinomial x 2 + 9 x − 22 and asked to find its factorization from the given options.
Finding the Factors To factor the trinomial x 2 + 9 x − 22 , we need to find two numbers that multiply to -22 and add up to 9. Let's list the factor pairs of -22:
(1, -22), (-1, 22), (2, -11), (-2, 11)
Checking the Sum Now, let's check which of these pairs adds up to 9:
1 + (-22) = -21 -1 + 22 = 21 2 + (-11) = -9 -2 + 11 = 9
The pair -2 and 11 satisfy the condition.
Writing the Factorization Therefore, the factorization of the trinomial is ( x − 2 ) ( x + 11 ) or equivalently ( x + 11 ) ( x − 2 ) .
Selecting the Correct Option Comparing this result with the given options, we see that option A, ( x + 11 ) ( x − 2 ) , matches our factorization.
Final Answer Thus, the correct factorization of the trinomial x 2 + 9 x − 22 is ( x + 11 ) ( x − 2 ) .
Examples
Factoring quadratic trinomials is a fundamental skill in algebra and has numerous applications in real life. For instance, imagine you are designing a rectangular garden and you know the area can be represented by the expression x 2 + 9 x − 22 , where x is a variable related to the dimensions. By factoring this expression into ( x + 11 ) ( x − 2 ) , you determine the possible lengths and widths of the garden. Knowing these factors helps you plan the layout and optimize the use of space, ensuring your garden fits perfectly in your yard.
The factorization of the trinomial x 2 + 9 x − 22 is ( x + 11 ) ( x − 2 ) . This is found by identifying two numbers that multiply to -22 and add up to 9, which are -2 and 11. Thus, the correct answer is option A: ( x + 11 ) ( x − 2 ) .
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