Factor Polynomial 89Ea18

1. **State the problem:** We want to factor the polynomial $$12x^3 - 15x^2 + 2$$. 2. **Look for common factors:** Check if there is a greatest common factor (GCF) among the terms. 3. The terms are $12x^3$, $-15x^2$, and $2$. The coefficients are $12$, $-15$, and $2$. The GCF of $12$, $15$, and $2$ is $1$, and the variable $x$ is not common to all terms (since $2$ has no $x$). 4. Since there is no common factor other than $1$, we cannot factor out a GCF. 5. **Try factoring by grouping or other methods:** Since this is a cubic polynomial with three terms, try to factor it as a product of a linear and a quadratic polynomial: $$12x^3 - 15x^2 + 2 = (ax + b)(cx^2 + dx + e)$$ 6. Expand the right side: $$ (ax + b)(cx^2 + dx + e) = acx^3 + adx^2 + aex + bcx^2 + bdx + be $$ Group like terms: $$ acx^3 + (ad + bc)x^2 + (ae + bd)x + be $$ 7. Match coefficients with the original polynomial: $$ ac = 12 $$ $$ ad + bc = -15 $$ $$ ae + bd = 0 $$ (since there is no $x$ term in the original) $$ be = 2 $$ 8. Try integer factors for $a$, $c$, $b$, $d$, $e$ that satisfy these equations. 9. After testing possible integer combinations, no integer factorization satisfies all conditions. 10. **Conclusion:** The polynomial $$12x^3 - 15x^2 + 2$$ does not factor nicely over the integers. Therefore, the polynomial is prime over the integers and cannot be factored further using integer coefficients.