Factor Rational Expression E3C2B5

1. **State the problem:** Factor the expression $$\frac{108p^2 - 153p - 54}{96p^2 + 104p + 24}$$ using grade 8 factoring techniques. 2. **Factor numerator:** - First, find the greatest common factor (GCF) of the numerator terms: 108, -153, and -54. - GCF is 9. - Factor out 9: $$108p^2 - 153p - 54 = 9(12p^2 - 17p - 6)$$ 3. **Factor denominator:** - Find GCF of denominator terms: 96, 104, and 24. - GCF is 8. - Factor out 8: $$96p^2 + 104p + 24 = 8(12p^2 + 13p + 3)$$ 4. **Factor the quadratic inside numerator:** - Factor $$12p^2 - 17p - 6$$. - Multiply $$a \times c = 12 \times (-6) = -72$$. - Find two numbers that multiply to -72 and add to -17: -24 and 3. - Rewrite middle term: $$12p^2 - 24p + 3p - 6$$ - Group terms: $$(12p^2 - 24p) + (3p - 6)$$ - Factor each group: $$12p(p - 2) + 3(p - 2)$$ - Factor out common binomial: $$(p - 2)(12p + 3)$$ 5. **Factor the quadratic inside denominator:** - Factor $$12p^2 + 13p + 3$$. - Multiply $$a \times c = 12 \times 3 = 36$$. - Find two numbers that multiply to 36 and add to 13: 9 and 4. - Rewrite middle term: $$12p^2 + 9p + 4p + 3$$ - Group terms: $$(12p^2 + 9p) + (4p + 3)$$ - Factor each group: $$3p(4p + 3) + 1(4p + 3)$$ - Factor out common binomial: $$(4p + 3)(3p + 1)$$ 6. **Rewrite entire expression:** $$\frac{9(p - 2)(12p + 3)}{8(4p + 3)(3p + 1)}$$ 7. **Simplify factors:** - Notice $$12p + 3 = 3(4p + 1)$$, but denominator has $$4p + 3$$ and $$3p + 1$$, which are different. - No common binomial factors to cancel. 8. **Final simplified form:** $$\frac{9(p - 2)(12p + 3)}{8(4p + 3)(3p + 1)}$$ This is the fully factored form using grade 8 techniques.