Find the greatest common factor of the
terms of the polynomial.

Write your answer as a constant times a
product of single variables raised to
exponents.

Ask by Ellis Goodwin. in the United States

Dec 09,2024

Question

Find the greatest common factor of the terms of the polynomial. $$ 10 a^{2} b^{7}+10 b^{10}+40 b^{5}+10 b $$ Write your answer as a constant times a product of single variables raised to exponents.

UpStudy AI · Accepted Answer

To find the greatest common factor (GCF) of the polynomial

$$
10 a^{2} b^{7} + 10 b^{10} + 40 b^{5} + 10 b,
$$

we will first identify the coefficients and the variable parts of each term.

1. **Coefficients**:
   - The coefficients of the terms are: 
     - $$10$$ from $$10 a^{2} b^{7}$$,
     - $$10$$ from $$10 b^{10}$$,
     - $$40$$ from $$40 b^{5}$$,
     - $$10$$ from $$10 b$$.

The GCF of the coefficients $$10, 10, 40, 10$$ is $$10$$.

2. **Variable Parts**:
   - The variable parts of the terms are:
     - $$a^{2} b^{7}$$,
     - $$b^{10}$$,
     - $$b^{5}$$,
     - $$b$$.

For the variable $$a$$, the minimum exponent present in the terms is $$0$$ (since $$b$$ does not have $$a$$).

For the variable $$b$$, the exponents are $$7, 10, 5, 1$$. The minimum exponent is $$1$$.

Therefore, the GCF for the variable parts is $$b^{1} = b$$.

3. **Combining the GCFs**:
   Now, we combine the GCF of the coefficients and the GCF of the variable parts:

$$
   \text{GCF} = 10 \cdot b^{1} = 10b.
   $$

Thus, the greatest common factor of the terms of the polynomial is

$$
\boxed{10b}.
$$
The greatest common factor is $$10b$$.

Find the greatest common factor of the
terms of the polynomial.

Write your answer as a constant times a
product of single variables raised to
exponents.

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Find the greatest common factor of the terms of the polynomial. Write your answer as a constant times a product of single variables raised to exponents.