Which of the following represents the factorization of the polynomial below?
[tex]$x^2+13 x+42$[/tex]
A. [tex]$(x+7)(x+6)$[/tex]
B. [tex]$(x+9)(x+4)$[/tex]
C. [tex]$(x+5)(x+8)$[/tex]
D. [tex]$(x+21)(x+21)$[/tex]
Answer (1)
We need to factor x 2 + 13 x + 42 .
Find two numbers that multiply to 42 and add up to 13. These numbers are 6 and 7.
Write the factorization as ( x + 6 ) ( x + 7 ) .
The correct factorization is ( x + 7 ) ( x + 6 ) .
Explanation
Understanding the Problem We are asked to factor the quadratic polynomial x 2 + 13 x + 42 . This means we need to find two binomials of the form ( x + a ) ( x + b ) such that when multiplied together, they equal the given polynomial.
Finding the Factors To factor the quadratic x 2 + 13 x + 42 , we need to find two numbers, a and b , such that their product is 42 (the constant term) and their sum is 13 (the coefficient of the x term).
Identifying the Correct Pair From the python calculation tool, we have the following pairs of factors of 42: (1, 42), (2, 21), (3, 14), (6, 7). Let's check which pair adds up to 13:
1 + 42 = 43
2 + 21 = 23
3 + 14 = 17
6 + 7 = 13
The pair (6, 7) satisfies the condition since 6 * 7 = 42 and 6 + 7 = 13.
Writing the Factorization Therefore, the factorization of the polynomial is ( x + 6 ) ( x + 7 ) .
Selecting the Correct Option Comparing our result with the given options, we see that option A, ( x + 7 ) ( x + 6 ) , matches our factorization.
Examples
Factoring quadratic polynomials is a fundamental skill in algebra and has many real-world applications. For example, engineers use factoring to design structures and predict their stability. Imagine you're designing a rectangular garden with an area represented by x 2 + 13 x + 42 . By factoring this expression into ( x + 6 ) ( x + 7 ) , you determine the dimensions of the garden to be ( x + 6 ) and ( x + 7 ) . This allows you to plan the layout efficiently and ensure you have enough space for your plants.
