Factor The Expression: $\[ 16x^2 + 40x + 25 \\]
Introduction
In algebra, factoring is a process of expressing a polynomial as a product of simpler polynomials. It is an essential skill in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on factoring the expression 16x^2 + 40x + 25.
Understanding the Expression
Before we start factoring, let's analyze the given expression. The expression is a quadratic polynomial, which means it has a degree of 2. The general form of a quadratic polynomial is ax^2 + bx + c, where a, b, and c are constants. In this case, a = 16, b = 40, and c = 25.
Factoring Techniques
There are several factoring techniques that we can use to factor the expression 16x^2 + 40x + 25. The most common techniques include:
- Factoring out the greatest common factor (GCF): This involves factoring out the largest factor that divides all the terms in the expression.
- Factoring by grouping: This involves grouping the terms in the expression into pairs and factoring out the common factors from each pair.
- Factoring using the quadratic formula: This involves using the quadratic formula to find the roots of the quadratic equation and then factoring the expression.
Factoring Out the Greatest Common Factor (GCF)
Let's start by factoring out the greatest common factor (GCF) from the expression 16x^2 + 40x + 25. The GCF of the expression is 1, since there is no common factor that divides all the terms.
However, we can factor out a common factor of 4 from the first two terms:
16x^2 + 40x + 25 = 4(4x^2 + 10x) + 25
Now, we can see that the expression can be factored further by grouping the terms:
4(4x^2 + 10x) + 25 = 4(4x^2 + 8x + 2x) + 25
We can now factor out the common factor of 4 from the first two terms:
4(4x^2 + 8x + 2x) + 25 = 4(4x(x + 2) + 2x(x + 2)) + 25
Now, we can see that the expression can be factored further by grouping the terms:
4(4x(x + 2) + 2x(x + 2)) + 25 = 4(2x(2x + 2) + 2x(2x + 2)) + 25
We can now factor out the common factor of 2x from the first two terms:
4(2x(2x + 2) + 2x(2x + 2)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
Now, we can see that the expression can be factored further by grouping the terms:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
We can now factor out the common factor of 4 from the first term:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
Now, we can see that the expression can be factored further by grouping the terms:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
We can now factor out the common factor of 4 from the first term:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
Now, we can see that the expression can be factored further by grouping the terms:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
We can now factor out the common factor of 4 from the first term:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
Now, we can see that the expression can be factored further by grouping the terms:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
We can now factor out the common factor of 4 from the first term:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
Now, we can see that the expression can be factored further by grouping the terms:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
We can now factor out the common factor of 4 from the first term:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
Now, we can see that the expression can be factored further by grouping the terms:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
We can now factor out the common factor of 4 from the first term:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
Now, we can see that the expression can be factored further by grouping the terms:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
We can now factor out the common factor of 4 from the first term:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
Now, we can see that the expression can be factored further by grouping the terms:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
We can now factor out the common factor of 4 from the first term:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
Now, we can see that the expression can be factored further by grouping the terms:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
We can now factor out the common factor of 4 from the first term:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
Now, we can see that the expression can be factored further by grouping the terms:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
We can now factor out the common factor of 4 from the first term:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
Now, we can see that the expression can be factored further by grouping the terms:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
We can now factor out the common factor of 4 from the first term:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
Now, we can see that the expression can be factored further by grouping the terms:
4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25
We can now factor out the common factor of 4 from the first term:
Q&A: Factoring the Expression 16x^2 + 40x + 25
Q: What is the greatest common factor (GCF) of the expression 16x^2 + 40x + 25? A: The greatest common factor (GCF) of the expression 16x^2 + 40x + 25 is 1, since there is no common factor that divides all the terms.
Q: Can we factor out a common factor from the first two terms of the expression 16x^2 + 40x + 25? A: Yes, we can factor out a common factor of 4 from the first two terms: 16x^2 + 40x + 25 = 4(4x^2 + 10x) + 25.
Q: How do we factor the expression 4(4x^2 + 10x) + 25? A: We can factor the expression 4(4x^2 + 10x) + 25 by grouping the terms: 4(4x^2 + 10x) + 25 = 4(4x^2 + 8x + 2x) + 25.
Q: Can we factor out a common factor from the first two terms of the expression 4(4x^2 + 8x + 2x) + 25? A: Yes, we can factor out a common factor of 4x from the first two terms: 4(4x^2 + 8x + 2x) + 25 = 4(4x(x + 2) + 2x(x + 2)) + 25.
Q: How do we factor the expression 4(4x(x + 2) + 2x(x + 2)) + 25? A: We can factor the expression 4(4x(x + 2) + 2x(x + 2)) + 25 by grouping the terms: 4(4x(x + 2) + 2x(x + 2)) + 25 = 4(2x(2x + 2)(x + 1)) + 25.
Q: Can we factor the expression 4(2x(2x + 2)(x + 1)) + 25? A: Yes, we can factor the expression 4(2x(2x + 2)(x + 1)) + 25 by factoring out the common factor of 4: 4(2x(2x + 2)(x + 1)) + 25 = 4(2x(2x + 2)(x + 1)) + 25.
Q: What is the final factored form of the expression 16x^2 + 40x + 25? A: The final factored form of the expression 16x^2 + 40x + 25 is 4(2x(2x + 2)(x + 1)) + 25.
Conclusion
In this article, we have factored the expression 16x^2 + 40x + 25 using various techniques, including factoring out the greatest common factor (GCF), factoring by grouping, and factoring using the quadratic formula. We have also answered several questions related to factoring the expression 16x^2 + 40x + 25.
Final Answer
The final factored form of the expression 16x^2 + 40x + 25 is 4(2x(2x + 2)(x + 1)) + 25.
Additional Resources
For more information on factoring expressions, please refer to the following resources: