To factor the given trinomial [tex]\( x^2 + 10x + 16 \)[/tex], we will follow a systematic approach. Here's a step-by-step solution:
### Step 1: Identify the Coefficients
The general form of a trinomial is given by:
[tex]\[ ax^2 + bx + c \][/tex]
For the given trinomial [tex]\( x^2 + 10x + 16 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 10 \)[/tex]
- [tex]\( c = 16 \)[/tex]
### Step 2: Find Two Numbers that Multiply to [tex]\( a \cdot c \)[/tex] and Add to [tex]\( b \)[/tex]
We need to find two numbers such that:
1. Their product is [tex]\( a \cdot c = 1 \cdot 16 = 16 \)[/tex]
2. Their sum is [tex]\( b = 10 \)[/tex]
Let's identify such a pair of numbers. After examining possible pairs, we find:
- 2 and 8
These two numbers fit our criteria because:
- [tex]\( 2 \cdot 8 = 16 \)[/tex] (the required product)
- [tex]\( 2 + 8 = 10 \)[/tex] (the required sum)
### Step 3: Rewrite the Middle Term Using the Pair of Numbers
We can rewrite the middle term [tex]\( 10x \)[/tex] as [tex]\( 2x + 8x \)[/tex]. Therefore:
[tex]\[ x^2 + 10x + 16 = x^2 + 2x + 8x + 16 \][/tex]
### Step 4: Factor by Grouping
Next, we group the terms to factor them step-by-step:
[tex]\[ x^2 + 2x + 8x + 16 \][/tex]
Group the terms:
[tex]\[ (x^2 + 2x) + (8x + 16) \][/tex]
Factor each group:
[tex]\[ x(x + 2) + 8(x + 2) \][/tex]
### Step 5: Factor Out the Common Binomial
Notice that both groups contain the common binomial [tex]\( (x + 2) \)[/tex]. We can factor this out:
[tex]\[ (x + 2)(x + 8) \][/tex]
### Conclusion
The factored form of the trinomial [tex]\( x^2 + 10x + 16 \)[/tex] is:
[tex]\[ (x + 2)(x + 8) \][/tex]
So, the pair of numbers that have a product of 16 and a sum of 10 is [tex]\( (2, 8) \)[/tex], and the factored form of the trinomial [tex]\( x^2 + 10x + 16 \)[/tex] is [tex]\( (x + 2)(x + 8) \)[/tex].
