Question
4x2 is the GCF of this polynomial.
20x2y + 56x3 – ?
Which could be the mystery term?
22x3
24x2y
26x2y
28y3
20x2y + 56x3 – ?
Which could be the mystery term?
22x3
24x2y
26x2y
28y3
Answers
GPT-4o mini
To find the mystery term in the polynomial \(20x^2y + 56x^3 - ?\) such that the greatest common factor (GCF) of the polynomial is \(4x^2\), we can start by identifying the terms that should be present in the polynomial.
The GCF \(4x^2\) means that each term in the polynomial can be expressed as a product involving \(4x^2\).
### Analysis of Each Term
1. **First Term**: \(20x^2y\)
- Factor \(20x^2y\) as \(4x^2 \cdot 5y\).
2. **Second Term**: \(56x^3\)
- Factor \(56x^3\) as \(4x^2 \cdot 14x\).
With these two terms, we can see that they both include \(4x^2\).
### Finding the Mystery Term
Let the mystery term be represented as \(4x^2 \cdot k\) for some expression \(k\).
The key is to ensure that the GCF remains \(4x^2\), meaning the additional term must also contain \(4x^2\) in its factorization.
Now let's analyze each of the provided choices to see if they can follow the GCF condition:
1. **\(22x^3\)**:
- Factorization: \(22x^3 = 2 \cdot 11x^3\) (does not contain \(4x^2\), so it can't be the mystery term).
2. **\(24x^2y\)**:
- Factorization: \(24x^2y = 4x^2 \cdot 6y\) (contains \(4x^2\), so it could be the mystery term).
3. **\(26x^2y\)**:
- Factorization: \(26x^2y = 2 \cdot 13x^2y\) (does not include \(4\) as a factor, so it can't be the mystery term).
4. **\(28y^3\)**:
- Factorization: \(28y^3 = 4 \cdot 7y^3\) (does not contain \(x^{2}\), hence it can't be the mystery term).
### Conclusion
The only term among the options that can serve as the mystery term while maintaining \(4x^2\) as the GCF is \(24x^2y\). Thus, the mystery term is:
**\(24x^2y\)**.
The GCF \(4x^2\) means that each term in the polynomial can be expressed as a product involving \(4x^2\).
### Analysis of Each Term
1. **First Term**: \(20x^2y\)
- Factor \(20x^2y\) as \(4x^2 \cdot 5y\).
2. **Second Term**: \(56x^3\)
- Factor \(56x^3\) as \(4x^2 \cdot 14x\).
With these two terms, we can see that they both include \(4x^2\).
### Finding the Mystery Term
Let the mystery term be represented as \(4x^2 \cdot k\) for some expression \(k\).
The key is to ensure that the GCF remains \(4x^2\), meaning the additional term must also contain \(4x^2\) in its factorization.
Now let's analyze each of the provided choices to see if they can follow the GCF condition:
1. **\(22x^3\)**:
- Factorization: \(22x^3 = 2 \cdot 11x^3\) (does not contain \(4x^2\), so it can't be the mystery term).
2. **\(24x^2y\)**:
- Factorization: \(24x^2y = 4x^2 \cdot 6y\) (contains \(4x^2\), so it could be the mystery term).
3. **\(26x^2y\)**:
- Factorization: \(26x^2y = 2 \cdot 13x^2y\) (does not include \(4\) as a factor, so it can't be the mystery term).
4. **\(28y^3\)**:
- Factorization: \(28y^3 = 4 \cdot 7y^3\) (does not contain \(x^{2}\), hence it can't be the mystery term).
### Conclusion
The only term among the options that can serve as the mystery term while maintaining \(4x^2\) as the GCF is \(24x^2y\). Thus, the mystery term is:
**\(24x^2y\)**.