The polynomial 2×3 − 5×2 + 4x − 10 is split into two groups, 2×3 + 4x and −5×2 − 10. The GCFs of each group is then factored out.
What is the common binomial factor between the two groups after their GCFs have been factored out?
a)2x + 5
b)2x − 5
c)x2 − 2
d)x2 + 2
The Correct Answer and Explanation is :
To find the common binomial factor after factoring the given polynomial (2x^3 – 5x^2 + 4x – 10) into two groups, we first need to identify and factor out the greatest common factor (GCF) from each group.
Step 1: Group the Terms
The polynomial can be split into two groups:
- (2x^3 + 4x)
- (-5x^2 – 10)
Step 2: Factor Out the GCF from Each Group
For the first group (2x^3 + 4x):
- The GCF of (2x^3) and (4x) is (2x).
- Factoring (2x) out gives:
[
2x(x^2 + 2)
]
For the second group (-5x^2 – 10):
- The GCF of (-5x^2) and (-10) is (-5).
- Factoring (-5) out gives:
[
-5(x^2 + 2)
]
Step 3: Combine the Factored Groups
Now, we can combine the results from both groups:
[
2x(x^2 + 2) – 5(x^2 + 2)
]
Notice that both terms contain the common binomial factor (x^2 + 2).
Step 4: Factor Out the Common Binomial Factor
Factoring out (x^2 + 2) from the entire expression yields:
[
(x^2 + 2)(2x – 5)
]
Conclusion
The common binomial factor after factoring the polynomial is:
[
x^2 + 2
]
Thus, the correct answer is (d) (x^2 + 2).
This method demonstrates the importance of grouping and factoring in polynomial expressions. By identifying GCFs, we can simplify complex expressions and reveal common factors that help in further analysis or solving equations. Understanding these techniques is fundamental in algebra and helps lay the groundwork for more advanced concepts, such as polynomial division and the Factor Theorem.