Factor Practice 67Ec56

1. **Problem:** Simplify $$\frac{-4u^2 + 4uv - v^2}{2u^2 + uv - v^2}$$ 2. **Factor numerator:** $$-4u^2 + 4uv - v^2 = -(4u^2 - 4uv + v^2)$$ Recognize perfect square trinomial: $$4u^2 - 4uv + v^2 = (2u - v)^2$$ So numerator is: $$-(2u - v)^2$$ 3. **Factor denominator:** $$2u^2 + uv - v^2$$ Find two numbers that multiply to $$2 \times (-1) = -2$$ and add to $$1$$ (coefficient of $$uv$$): these are $$2$$ and $$-1$$. Rewrite middle term: $$2u^2 + 2uv - uv - v^2$$ Group: $$(2u^2 + 2uv) - (uv + v^2)$$ Factor each group: $$2u(u + v) - v(u + v)$$ Factor out common binomial: $$(2u - v)(u + v)$$ 4. **Simplify expression:** $$\frac{-(2u - v)^2}{(2u - v)(u + v)}$$ Cancel common factor $$2u - v$$: $$\frac{-(\cancel{2u - v})(2u - v)}{\cancel{2u - v}(u + v)} = \frac{-(2u - v)}{u + v}$$ --- 1. **Problem:** Simplify $$\frac{x - 7}{(6x - 41)x - 7}$$ 2. **Simplify denominator:** $$ (6x - 41)x - 7 = 6x^2 - 41x - 7$$ 3. **Factor denominator:** Find two numbers that multiply to $$6 \times (-7) = -42$$ and add to $$-41$$. These numbers are $$-42$$ and $$1$$. Rewrite middle term: $$6x^2 - 42x + x - 7$$ Group: $$(6x^2 - 42x) + (x - 7)$$ Factor each group: $$6x(x - 7) + 1(x - 7)$$ Factor out common binomial: $$(6x + 1)(x - 7)$$ 4. **Simplify expression:** $$\frac{x - 7}{(6x + 1)(x - 7)}$$ Cancel common factor $$x - 7$$: $$\frac{\cancel{x - 7}}{(6x + 1)\cancel{(x - 7)}} = \frac{1}{6x + 1}$$ --- 1. **Problem:** Simplify $$\frac{x(x^3 + 7x^2) + 2x^2(x + 9)}{x^4 - 2(x^3 + 7x^2) - x^2}$$ 2. **Simplify numerator:** $$x(x^3 + 7x^2) + 2x^2(x + 9) = x^4 + 7x^3 + 2x^3 + 18x^2 = x^4 + 9x^3 + 18x^2$$ 3. **Simplify denominator:** $$x^4 - 2(x^3 + 7x^2) - x^2 = x^4 - 2x^3 - 14x^2 - x^2 = x^4 - 2x^3 - 15x^2$$ 4. **Factor numerator:** Factor out $$x^2$$: $$x^2(x^2 + 9x + 18)$$ Factor quadratic: $$x^2(x + 6)(x + 3)$$ 5. **Factor denominator:** Factor out $$x^2$$: $$x^2(x^2 - 2x - 15)$$ Factor quadratic: $$x^2(x - 5)(x + 3)$$ 6. **Simplify expression:** $$\frac{x^2(x + 6)(x + 3)}{x^2(x - 5)(x + 3)}$$ Cancel common factors $$x^2$$ and $$x + 3$$: $$\frac{\cancel{x^2}(x + 6)\cancel{(x + 3)}}{\cancel{x^2}(x - 5)\cancel{(x + 3)}} = \frac{x + 6}{x - 5}$$ --- 1. **Problem:** Simplify $$\frac{-108p^2 - 153p - 54}{96p^2 + 104p + 24}$$ 2. **Factor numerator:** Factor out $$-3$$: $$-3(36p^2 + 51p + 18)$$ Factor quadratic inside: Find two numbers that multiply to $$36 \times 18 = 648$$ and add to $$51$$. These are $$27$$ and $$24$$. Rewrite middle term: $$36p^2 + 27p + 24p + 18$$ Group: $$(36p^2 + 27p) + (24p + 18)$$ Factor each group: $$9p(4p + 3) + 6(4p + 3)$$ Factor out common binomial: $$(9p + 6)(4p + 3)$$ So numerator: $$-3(9p + 6)(4p + 3)$$ 3. **Factor denominator:** Factor out $$8$$: $$8(12p^2 + 13p + 3)$$ Factor quadratic inside: Find two numbers that multiply to $$12 \times 3 = 36$$ and add to $$13$$. These are $$9$$ and $$4$$. Rewrite middle term: $$12p^2 + 9p + 4p + 3$$ Group: $$(12p^2 + 9p) + (4p + 3)$$ Factor each group: $$3p(4p + 3) + 1(4p + 3)$$ Factor out common binomial: $$(3p + 1)(4p + 3)$$ So denominator: $$8(3p + 1)(4p + 3)$$ 4. **Simplify expression:** $$\frac{-3(9p + 6)(4p + 3)}{8(3p + 1)(4p + 3)}$$ Cancel common factor $$4p + 3$$: $$\frac{-3(9p + 6)\cancel{(4p + 3)}}{8(3p + 1)\cancel{(4p + 3)}}$$ Factor $$9p + 6$$: $$3(3p + 2)$$ So numerator becomes: $$-3 \times 3 (3p + 2) = -9(3p + 2)$$ 5. **Final simplified form:** $$\frac{-9(3p + 2)}{8(3p + 1)}$$