Question
Factor: Se l by group: \( x^{2}+4 x+3 x+20 \) \( 2 x^{2}+8 x^{2}-2 x \)
Ask by Sherman Dickson. in South Africa
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
1. \( x^{2} + 4x + 3x + 20 \) cannot be factored over the integers.
2. \( 2x^{2} + 8x^{2} - 2x = 2x(5x - 1) \).
Solution
Sure, let's factor each of the given polynomial expressions by grouping.
### 1. \( x^{2} + 4x + 3x + 20 \)
**Step 1: Combine like terms.**
\[
x^{2} + 4x + 3x + 20 = x^{2} + 7x + 20
\]
**Step 2: Attempt to factor by grouping.**
We look for two numbers that multiply to \( 20 \) (the constant term) and add up to \( 7 \) (the coefficient of \( x \)).
However, the pairs of factors of 20 are:
- \( 1 \) and \( 20 \) (sum = \( 21 \))
- \( 2 \) and \( 10 \) (sum = \( 12 \))
- \( 4 \) and \( 5 \) (sum = \( 9 \))
None of these pairs add up to \( 7 \).
**Conclusion:**
The polynomial \( x^{2} + 7x + 20 \) **cannot be factored** using integer coefficients.
---
### 2. \( 2x^{2} + 8x^{2} - 2x \)
**Step 1: Combine like terms.**
\[
2x^{2} + 8x^{2} - 2x = 10x^{2} - 2x
\]
**Step 2: Factor out the greatest common factor (GCF).**
The GCF of \( 10x^{2} \) and \( -2x \) is \( 2x \).
\[
10x^{2} - 2x = 2x(5x - 1)
\]
**Conclusion:**
The polynomial factors as \( 2x(5x - 1) \).
---
**Final Summary:**
1. \( x^{2} + 4x + 3x + 20 = x^{2} + 7x + 20 \) **cannot be factored** over the integers.
2. \( 2x^{2} + 8x^{2} - 2x = 2x(5x - 1) \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
Let's break it down step by step! First, for \( x^{2} + 4x + 3x + 20 \), you can combine like terms to get \( x^{2} + 7x + 20 \). Next, to factor by grouping, we can rewrite this as \( x^{2} + 7x + 20 = (x^2 + 5x) + (2x + 20) \). Now we can factor out common terms: \( x(x + 5) + 2(x + 5) = (x + 5)(x + 2) \). Now, for \( 2x^{2} + 8x^{2} - 2x \), combine the like terms to get \( 10x^{2} - 2x \). Factoring out the common factor of \( 2x \), we have \( 2x(5x - 1) \). In summary, the factors are \( (x + 5)(x + 2) \) from the first expression and \( 2x(5x - 1) \) from the second one.
