The polynomial 2x^(3)-5x^(2)+4x-10 is split into two groups, 2x^(3)+4x and -5x^(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?
The Correct Answer and Explanation is :
To factor the polynomial (2x^3 – 5x^2 + 4x – 10) by grouping, we first split it into two groups:
- First Group: (2x^3 + 4x)
- Second Group: (-5x^2 – 10)
Step 1: Factor out the Greatest Common Factor (GCF) from each group.
- First Group: (2x^3 + 4x)
- The GCF of (2x^3) and (4x) is (2x).
- Factoring out (2x):
[
2x(x^2 + 2)
] - Second Group: (-5x^2 – 10)
- The GCF of (-5x^2) and (-10) is (-5).
- Factoring out (-5):
[
-5(x^2 + 2)
]
Step 2: Combine the factored expressions.
After factoring out the GCFs, we have:
[
2x(x^2 + 2) – 5(x^2 + 2)
]
Step 3: Factor out the common binomial factor.
Both terms contain the binomial factor ((x^2 + 2)). Factoring this out:
[
(x^2 + 2)(2x – 5)
]
Conclusion:
The common binomial factor between the two groups after factoring out their GCFs is ((x^2 + 2)). Therefore, the factored form of the polynomial (2x^3 – 5x^2 + 4x – 10) is:
[
(x^2 + 2)(2x – 5)
]
Explanation:
Factoring by grouping is a method used to factor polynomials with four terms. The process involves:
- Grouping terms: Divide the polynomial into two groups that can each be factored.
- Factoring each group: Find and factor out the GCF from each group.
- Factoring out the common binomial: If both groups share a common binomial factor, factor it out.
In this case, after factoring out the GCFs from each group, we identified the common binomial factor ((x^2 + 2)) and factored it out, resulting in the final factored form ((x^2 + 2)(2x – 5)).