Find the best solutions to your problems with the help of IDNLearn.com's expert users. Our experts are available to provide accurate, comprehensive answers to help you make informed decisions about any topic or issue you encounter.
Sagot :
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].
### 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].
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.