Question

Factor.
100 x squared plus 20 x plus 1
Question content area bottom
Part 1
100 x squared plus 20 x plus 1equals
  
enter your response here ​(Factor completely.)

Answer 1

To factor the expression \(100x^2 + 20x + 1\), we can look for two numbers that multiply to give the product of the leading coefficient (100) and the constant term (1), which is \(100 \cdot 1 = 100\), and also add up to the middle coefficient (20).

The expression can be factored as follows:

1. **Identify the coefficients**:
- \(a = 100\)
- \(b = 20\)
- \(c = 1\)

2. **Calculate the product**:
- \(a \cdot c = 100 \cdot 1 = 100\)

3. **Find two numbers that multiply to 100 and add to 20**:
- The numbers are \(10\) and \(10\) because \(10 \cdot 10 = 100\) and \(10 + 10 = 20\).

4. **Rewrite the quadratic**:
\[
100x^2 + 10x + 10x + 1
\]

5. **Group the terms**:
\[
(100x^2 + 10x) + (10x + 1)
\]

6. **Factor by grouping**:
\[
10x(10x + 1) + 1(10x + 1)
\]

7. **Factor out the common binomial factor**:
\[
(10x + 1)(10x + 1) \text{ or } (10x + 1)^2
\]

Therefore, the completely factored form of \(100x^2 + 20x + 1\) is:
\[
(10x + 1)^2
\]

So, the final answer is:
\[
100x^2 + 20x + 1 = (10x + 1)^2
\]