Question
Factor the GCF out of the polynomial below: \( 6 x^{5}+9 x^{3}+15 x^{2} \)
Ask by Barrett Wagner. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The GCF is 3x², and the factored form is \( 3x^{2}(2x^{3} + 3x + 5) \).
Solution
To factor the greatest common factor (GCF) out of 6x⁵ + 9x³ + 15x², follow these steps:
1. Identify the GCF of the coefficients (6, 9, 15). The largest number that divides all of them is 3.
2. Identify the lowest power of x present in each term. The smallest exponent is 2 (x²).
Thus, the GCF is 3x².
Now, factor 3x² out of each term:
6x⁵ ÷ 3x² = 2x³
9x³ ÷ 3x² = 3x
15x² ÷ 3x² = 5
So, the expression factors as:
3x²(2x³ + 3x + 5)
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Simplify this solution The Deep Dive
To factor out the greatest common factor (GCF) from the polynomial \( 6x^{5} + 9x^{3} + 15x^{2} \), we first identify the GCF of the coefficients: 6, 9, and 15. The GCF of these numbers is 3. Next, we observe the variable part. The smallest power of \( x \) present in all terms is \( x^{2} \). Therefore, the overall GCF is \( 3x^{2} \). Now we can factor it out: \[ 6x^{5} + 9x^{3} + 15x^{2} = 3x^{2}(2x^{3} + 3x + 5) \] So the factored form of the polynomial is \( 3x^{2}(2x^{3} + 3x + 5) \).
