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20) \( 3 x^{3}+15 x^{2}+2 x+10 \)

Ask by Curry Hilton. in the United States
Mar 12,2025

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The expression \( 3x^{3} + 15x^{2} + 2x + 10 \) factors into \( (x + 5)(3x^{2} + 2) \).

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To factor the polynomial \( 3x^{3}+15x^{2}+2x+10 \), first, notice that we can group the terms. Grouping gives us \( (3x^{3} + 15x^{2}) + (2x + 10) \). From the first group, factor out \( 3x^{2} \), yielding \( 3x^{2}(x + 5) \). For the second group, factor out \( 2 \), resulting in \( 2(x + 5) \). Now, we have \( 3x^{2}(x + 5) + 2(x + 5) \). We can factor out the \( (x + 5) \) term, leaving us with \( (x + 5)(3x^{2} + 2) \). Thus, the complete factorization is \( (x + 5)(3x^{2} + 2) \). The factors \( (3x^{2} + 2) \) don't yield real roots (as its discriminant is negative), meaning it doesn't factor further over the real numbers. This polynomial is a nice example of how grouping can simplify the factoring process!

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