the given polynomial by finding the greatest common monomial factor for the negative of the greatest $$ -7 y+49 x y^{2}+49 x^{2} $$

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UpStudy AI · Accepted Answer

To find the greatest common monomial factor (GCMF) of the polynomial $$-7y + 49xy^2 + 49x^2$$, we will follow these steps:

1. **Identify the coefficients and variables in each term:**
   - The terms are:
     - $$-7y$$
     - $$49xy^2$$
     - $$49x^2$$

2. **Find the GCMF of the coefficients:**
   - The coefficients are $$-7$$, $$49$$, and $$49$$.
   - The GCMF of $$-7$$ and $$49$$ is $$7$$ (since $$-7$$ is negative, we will take the positive value for the GCMF).

3. **Identify the common variables:**
   - The variables in each term are:
     - $$-7y$$ has $$y^1$$
     - $$49xy^2$$ has $$x^1y^2$$
     - $$49x^2$$ has $$x^2$$
   - The common variable is $$y$$, and the lowest power of $$y$$ present in all terms is $$y^0$$ (since $$49x^2$$ does not have $$y$$).

4. **Combine the GCMF of the coefficients and the common variables:**
   - The GCMF is $$7$$.

5. **Factor out the GCMF from the polynomial:**
   - We will factor out $$-7$$ (the negative of the GCMF) from the polynomial:
   $$
   -7(y - 7xy + 7x^2)
   $$

Thus, the polynomial factored by the negative of the greatest common monomial factor is:
$$
-7(y - 7xy + 7x^2)
$$
The greatest common monomial factor is $$-7$$, and factoring it out gives $$-7(y - 7xy + 7x^2)$$.

the given polynomial by finding the greatest common monomial factor for the negative of the greatest \[ -7 y+49 x y^{2}+49 x^{2} \]

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