Question
Factor this polynomial. Then select the correct factors from the options below. (You will select two factors) \( 15 x^{3}+40 x^{2}+3 x+8 \)
Ask by Collins Estrada. in the United States
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The factors are (3x + 8) and (5x² + 1).
Solution
To factor 15x³ + 40x² + 3x + 8, we can use grouping:
1. Group the polynomial into two pairs:
(15x³ + 40x²) + (3x + 8)
2. Factor out the common factors from each group:
From 15x³ + 40x², factor out 5x²:
5x²(3x + 8)
From 3x + 8, factor out 1:
1(3x + 8)
3. Now the polynomial becomes:
5x²(3x + 8) + 1(3x + 8)
4. Factor out the common binomial (3x + 8):
(3x + 8)(5x² + 1)
Thus, the factors you should select are (3x + 8) and (5x² + 1).
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Simplify this solution Beyond the Answer
To factor the polynomial \( 15 x^{3}+40 x^{2}+3 x+8 \), we can use techniques such as grouping. We can group the first two terms and the last two terms: \[ (15x^3 + 40x^2) + (3x + 8) \] Factoring out common elements in each group gives: \[ 5x^2(3x + 8) + 1(3x + 8) \] Next, we can factor out the common term \( (3x + 8) \): \[ (3x + 8)(5x^2 + 1) \] So, the correct factors are \( (3x + 8) \) and \( (5x^2 + 1) \). 5x^2 + 1 and 3x + 8 would be the two selected factors.
