Question

which term can be used in the blank of \\(36x^3 - 22x^2 - \underline{\quad}\\) so the greatest common factor of the resulting polynomial is \\(2x\\)? select two options.

2
4xy
12x
24
44y

Explanation:

Analyze the given terms and the required GCF

The given terms are \(36x^3\) and \(-22x^2\).
We want the greatest common factor (GCF) of the polynomial \(36x^3 - 22x^2 - \text{blank}\) to be \(2x\).
For the GCF of the polynomial to be exactly \(2x\):

1. Every term in the polynomial must be divisible by \(2x\).
2. The GCF of the three terms cannot be any larger than \(2x\) (e.g., it cannot be \(4x\), \(2x^2\), etc.).

Test divisibility by \(2x\) for each option

For a term to be divisible by \(2x\), its coefficient must be divisible by \(2\) and it must contain at least \(x^1\) (so it must have a factor of \(x\)).

• Option 1: \(2\) is not divisible by \(x\).
• Option 2: \(4xy\) is divisible by \(2x\) since \(\frac{4xy}{2x} = 2y\).
• Option 3: \(12x\) is divisible by \(2x\) since \(\frac{12x}{2x} = 6\).
• Option 4: \(24\) is not divisible by \(x\).
• Option 5: \(44y\) is not divisible by \(x\).

Thus, only \(4xy\) and \(12x\) are divisible by \(2x\).

Verify the GCF of the resulting polynomials

• If the blank is \(4xy\), the polynomial is \(36x^3 - 22x^2 - 4xy\).

The GCF of the coefficients \(\{36, -22, -4\}\) is \(2\).
The GCF of the variable parts \(\{x^3, x^2, xy\}\) is \(x\).
Thus, the GCF is \(2x\). This option is correct.

• If the blank is \(12x\), the polynomial is \(36x^3 - 22x^2 - 12x\).

The GCF of the coefficients \(\{36, -22, -12\}\) is \(2\).
The GCF of the variable parts \(\{x^3, x^2, x\}\) is \(x\).
Thus, the GCF is \(2x\). This option is correct.

Answer:

• 2
• 4xy (Correct answer)
• 12x (Correct answer)
• 24
• 44y