Certainly! Let's solve the problem step by step.
### Consider the trinomial [tex]\( x^2 + 10x + 16 \)[/tex]
We need to find the factored form of this trinomial. To do this, we look for two numbers that satisfy the following conditions:
1. Product: The product of the two numbers should equal [tex]\( a \times c \)[/tex].
2. Sum: The sum of the two numbers should equal [tex]\( b \)[/tex].
For the trinomial of the form [tex]\( ax^2 + bx + c \)[/tex]:
- Here, [tex]\( a = 1 \)[/tex]
- [tex]\( b = 10 \)[/tex]
- [tex]\( c = 16 \)[/tex]
So, we need to find two numbers, [tex]\( p \)[/tex] and [tex]\( q \)[/tex], such that:
- [tex]\( p \times q = a \times c = 1 \times 16 = 16 \)[/tex]
- [tex]\( p + q = b = 10 \)[/tex]
### Finding the Pair of Numbers
We need to find pairs of numbers that multiply to 16 and add up to 10. By examining various pairs, we find that:
- [tex]\( 2 \times 8 = 16 \)[/tex]
- [tex]\( 2 + 8 = 10 \)[/tex]
Another pair is:
- [tex]\( 8 \times 2 = 16 \)[/tex]
- [tex]\( 8 + 2 = 10 \)[/tex]
Among these pairs, both pairs [tex]\((2, 8)\)[/tex] and [tex]\((8, 2)\)[/tex] meet the required conditions.
This means the pairs of numbers that have a product of 16 and a sum of 10 are [tex]\((2, 8)\)[/tex] or [tex]\((8, 2)\)[/tex].
### Factoring the Trinomial
Using the numbers [tex]\( p = 2 \)[/tex] and [tex]\( q = 8 \)[/tex], we can factor the trinomial. The trinomial can be written as:
[tex]\[ x^2 + 10x + 16 = (x + 2)(x + 8) \][/tex]
Thus, the factored form of the trinomial [tex]\( x^2 + 10x + 16 \)[/tex] is:
[tex]\[ (x + 2)(x + 8) \][/tex]
### Final Answer
- The pair of numbers that have a product of [tex]\( ac \)[/tex] (which is [tex]\( 16 \)[/tex]) and a sum of [tex]\( b \)[/tex] (which is [tex]\( 10 \)[/tex]) are [tex]\( (2, 8) \)[/tex] and [tex]\( (8, 2) \)[/tex].
- The factored form of the trinomial [tex]\( x^2 + 10x + 16 \)[/tex] is [tex]\( (x + 2)(x + 8) \)[/tex].
