Question
\( 1 \leftarrow \quad \) Factor out the greatest common factor from the following polynomial. \[ 7 y^{5}-28 x y^{6} \]
Ask by Lane Best. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The greatest common factor is \(7y^5\), and factoring it out gives \(7y^5(1 - 4xy)\).
Solution
1. Identify the greatest common factor (GCF) of the terms \( 7y^5 \) and \(-28xy^6 \).
- The coefficient \(7\) is a factor of both \(7\) and \(-28\).
- The variable \( y \) appears as \( y^5 \) in the first term and \( y^6 \) in the second term; the highest power that divides both is \( y^5 \).
- The variable \( x \) appears only in the second term, so it is not part of the GCF.
2. The GCF is \( 7y^5 \).
3. Factor \( 7y^5 \) out of each term:
\[
7y^5 - 28xy^6 = 7y^5\left( \frac{7y^5}{7y^5} - \frac{28xy^6}{7y^5} \right)
\]
Simplify each term inside the parentheses:
- First term: \( \frac{7y^5}{7y^5} = 1 \).
- Second term: \( \frac{-28xy^6}{7y^5} = -4xy \) (since \( \frac{28}{7} = 4 \) and \( \frac{y^6}{y^5} = y \)).
4. The factored form is:
\[
7y^5(1 - 4xy)
\]
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Simplify this solution Extra Insights
To factor out the greatest common factor (GCF) from the polynomial \(7y^5 - 28xy^6\), we start by identifying the coefficients and variables. The coefficients are 7 and -28; their GCF is 7. For the variable parts, \(y^5\) and \(y^6\), the GCF is \(y^5\). Thus, the GCF of the entire expression is \(7y^5\). We can factor this out: \[ 7y^5(1 - 4xy) \] So, the factored form of the polynomial is \(7y^5(1 - 4xy)\). Now, let's have a little fun with this: if polynomials were chefs, factoring out the GCF would be like a sous chef preparing the ingredients—getting rid of what's redundant to make the dish, or the equation, shine!
