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 Bowen Owen. in the United States

Nov 14,2024

Question

Find the greatest common factor of the terms of the polynomial. $$ 7 r^{10}+12 r^{7} s+4 r^{4}-10 s^{7} $$ 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 terms of the polynomial

$$
7 r^{10} + 12 r^{7} s + 4 r^{4} - 10 s^{7},
$$

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

The terms of the polynomial are:
1. $$7 r^{10}$$
2. $$12 r^{7} s$$
3. $$4 r^{4}$$
4. $$-10 s^{7}$$

### Step 1: Find the GCF of the coefficients
The coefficients are $$7$$, $$12$$, $$4$$, and $$-10$$. We will find the GCF of these numbers.

- The prime factorization of $$7$$ is $$7$$.
- The prime factorization of $$12$$ is $$2^2 \cdot 3$$.
- The prime factorization of $$4$$ is $$2^2$$.
- The prime factorization of $$-10$$ is $$-1 \cdot 2 \cdot 5$$.

The GCF of the coefficients $$7$$, $$12$$, $$4$$, and $$-10$$ is $$1$$ since there are no common prime factors among all coefficients.

### Step 2: Find the GCF of the variable parts
Next, we will find the GCF of the variable parts of each term.

- The term $$7 r^{10}$$ has $$r^{10}$$.
- The term $$12 r^{7} s$$ has $$r^{7} s$$.
- The term $$4 r^{4}$$ has $$r^{4}$$.
- The term $$-10 s^{7}$$ has $$s^{7}$$.

Now, we will find the GCF for the variable $$r$$:
- The powers of $$r$$ are $$10$$, $$7$$, and $$4$$. The minimum power is $$4$$, so the GCF for $$r$$ is $$r^{4}$$.

Next, we will find the GCF for the variable $$s$$:
- The powers of $$s$$ are $$1$$ (from $$12 r^{7} s$$) and $$7$$ (from $$-10 s^{7}$$). The minimum power is $$0$$ (since $$s$$ does not appear in $$7 r^{10}$$ and $$4 r^{4}$$), so the GCF for $$s$$ is $$s^{0} = 1$$.

### Step 3: Combine the GCF of coefficients and variables
The overall GCF of the polynomial is the product of the GCF of the coefficients and the GCF of the variable parts:

$$
\text{GCF} = 1 \cdot r^{4} \cdot 1 = r^{4}.
$$

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

$$
\boxed{r^{4}}.
$$
The greatest common factor of the polynomial is $$ r^{4} $$.

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.