Question
Factor out the greatest common factor from the following polynomial. \[ 33 x^{5} y^{5}-22 x^{4} y^{5} \]
Ask by Schneider Bowen. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The factored form of the polynomial is \(11x^4y^5(3x - 2)\).
Solution
1. Identify the greatest common factor (GCF) of the coefficients:
The coefficients are \(33\) and \(-22\). Their GCF is \(11\).
2. Determine the GCF for the variable \(x\):
The terms contain \(x^5\) and \(x^4\). The smallest power is \(x^4\).
3. Determine the GCF for the variable \(y\):
Both terms have \(y^5\).
4. The overall GCF is:
\[
11x^4y^5
\]
5. Factor \(11x^4y^5\) out of each term:
\[
33x^5y^5-22x^4y^5 = 11x^4y^5\left(\frac{33x^5y^5}{11x^4y^5} - \frac{22x^4y^5}{11x^4y^5}\right)
\]
6. Simplify each fraction:
\[
\frac{33x^5y^5}{11x^4y^5} = 3x \quad \text{and} \quad \frac{22x^4y^5}{11x^4y^5} = 2
\]
7. Write the factored form:
\[
33x^5y^5-22x^4y^5 = 11x^4y^5 (3x-2)
\]
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Simplify this solution Bonus Knowledge
To factor out the greatest common factor (GCF) from the polynomial \( 33 x^{5} y^{5} - 22 x^{4} y^{5} \), we first determine the GCF of the coefficients and the variables. The coefficients are 33 and 22. The GCF of 33 and 22 is 11. Next, we look at the variable parts: - For \( x^5 \) and \( x^4 \), the GCF is \( x^4 \) (the lowest power). - For \( y^5 \) in both terms, since both terms have \( y^5 \), the GCF is \( y^5 \). Combining these, the overall GCF is \( 11 x^4 y^5 \). Now we factor this out from the polynomial: \[ 33 x^{5} y^{5} - 22 x^{4} y^{5} = 11 x^{4} y^{5} (3x - 2) \] So the factored form of the polynomial is: \[ 11 x^{4} y^{5} (3x - 2) \]
