1. **Problem statement:** Factor the following expressions from problems 7 to 11, parts а to к. --- ### 7. а) $x^5 - x^3 + x^2 - 1$ 2. Group terms: $(x^5 - x^3) + (x^2 - 1)$ 3. Factor each group: $$x^3(x^2 - 1) + (x^2 - 1)$$ 4. Factor out common factor $(x^2 - 1)$: $$ (x^2 - 1)(x^3 + 1) $$ 5. Recognize difference of squares and sum of cubes: $$ (x - 1)(x + 1)(x + 1)(x^2 - x + 1) $$ 6. Final factorization: $$ (x - 1)(x + 1)^2(x^2 - x + 1) $$ --- ### 7. в) $a^3 - 8 + 6a^2 - 12a$ 2. Rearrange terms: $$ a^3 + 6a^2 - 12a - 8 $$ 3. Group: $$ (a^3 + 6a^2) - (12a + 8) $$ 4. Factor each group: $$ a^2(a + 6) - 4(3a + 2) $$ 5. No common binomial factor, try rearranging: $$ (a^3 - 12a) + (6a^2 - 8) = a(a^2 - 12) + 2(3a^2 - 4) $$ 6. No simple factorization, try factoring as cubic: $$ a^3 + 6a^2 - 12a - 8 = (a + 2)^2(a - 2) $$ Check by expansion: $$(a + 2)^2(a - 2) = (a^2 + 4a + 4)(a - 2) = a^3 + 2a^2 - 4a - 8$$ Mismatch, so try grouping differently. 7. Factor by grouping: $$ (a^3 - 12a) + (6a^2 - 8) = a(a^2 - 12) + 2(3a^2 - 4) $$ No common factor, so try polynomial division or synthetic division. 8. Try root $a=2$: $$2^3 + 6(2)^2 - 12(2) - 8 = 8 + 24 - 24 - 8 = 0$$ So $(a - 2)$ is a factor. 9. Divide polynomial by $(a - 2)$: $$a^3 + 6a^2 - 12a - 8 = (a - 2)(a^2 + 8a + 4)$$ 10. Final factorization: $$ (a - 2)(a^2 + 8a + 4) $$ --- ### 7. д) $a^4 + a^3 + a + 1$ 2. Group terms: $$ (a^4 + a^3) + (a + 1) $$ 3. Factor each group: $$ a^3(a + 1) + 1(a + 1) $$ 4. Factor out common binomial: $$ (a + 1)(a^3 + 1) $$ 5. Recognize sum of cubes: $$ (a + 1)(a + 1)(a^2 - a + 1) = (a + 1)^2(a^2 - a + 1) $$ --- ### 7. ж) $a^3 + a^2b - ab^2 - b^3$ 2. Group terms: $$ (a^3 + a^2b) - (ab^2 + b^3) $$ 3. Factor each group: $$ a^2(a + b) - b^2(a + b) $$ 4. Factor out common binomial: $$ (a + b)(a^2 - b^2) $$ 5. Recognize difference of squares: $$ (a + b)(a - b)(a + b) = (a + b)^2(a - b) $$ --- ### 7. б) $m^5 + m^3 - m^2 - 1$ 2. Group terms: $$ (m^5 + m^3) - (m^2 + 1) $$ 3. Factor each group: $$ m^3(m^2 + 1) - 1(m^2 + 1) $$ 4. Factor out common binomial: $$ (m^2 + 1)(m^3 - 1) $$ 5. Recognize difference of cubes: $$ (m^2 + 1)(m - 1)(m^2 + m + 1) $$ --- ### 7. г) $p^3 + 8 + 6p^2 + 12p$ 2. Rearrange terms: $$ p^3 + 6p^2 + 12p + 8 $$ 3. Group: $$ (p^3 + 6p^2) + (12p + 8) $$ 4. Factor each group: $$ p^2(p + 6) + 4(3p + 2) $$ 5. No common binomial factor, try rearranging: $$ (p^3 + 12p) + (6p^2 + 8) = p(p^2 + 12) + 2(3p^2 + 4) $$ No common factor, try root $p = -2$: $$ (-2)^3 + 6(-2)^2 + 12(-2) + 8 = -8 + 24 - 24 + 8 = 0 $$ 6. So $(p + 2)$ is a factor. 7. Divide polynomial by $(p + 2)$: $$ p^3 + 6p^2 + 12p + 8 = (p + 2)(p^2 + 4p + 4) $$ 8. Recognize perfect square: $$ (p + 2)(p + 2)^2 = (p + 2)^3 $$ --- ### 7. е) $x^4 + x^3 - x - 1$ 2. Group terms: $$ (x^4 + x^3) - (x + 1) $$ 3. Factor each group: $$ x^3(x + 1) - 1(x + 1) $$ 4. Factor out common binomial: $$ (x + 1)(x^3 - 1) $$ 5. Recognize difference of cubes: $$ (x + 1)(x - 1)(x^2 + x + 1) $$ --- ### 7. з) $x^3 - x^2y - xy^2 + y^3$ 2. Group terms: $$ (x^3 - x^2y) - (xy^2 - y^3) $$ 3. Factor each group: $$ x^2(x - y) - y^2(x - y) $$ 4. Factor out common binomial: $$ (x - y)(x^2 - y^2) $$ 5. Recognize difference of squares: $$ (x - y)(x - y)(x + y) = (x - y)^2(x + y) $$ --- ### 8. а) $m^4 - n^4$ 2. Recognize difference of squares: $$ (m^2 - n^2)(m^2 + n^2) $$ 3. Further factor difference of squares: $$ (m - n)(m + n)(m^2 + n^2) $$ --- ### 8. в) $x^4 + x^3 + x + 1$ 2. Group terms: $$ (x^4 + x^3) + (x + 1) $$ 3. Factor each group: $$ x^3(x + 1) + 1(x + 1) $$ 4. Factor out common binomial: $$ (x + 1)(x^3 + 1) $$ 5. Recognize sum of cubes: $$ (x + 1)(x + 1)(x^2 - x + 1) = (x + 1)^2(x^2 - x + 1) $$ --- ### 8. д) $(a + b)^3 - (a - b)^3$ 2. Use difference of cubes formula: $$ (a + b - (a - b))((a + b)^2 + (a + b)(a - b) + (a - b)^2) $$ 3. Simplify first factor: $$ (a + b - a + b) = 2b $$ 4. Expand second factor: $$ (a + b)^2 + (a + b)(a - b) + (a - b)^2 = (a^2 + 2ab + b^2) + (a^2 - b^2) + (a^2 - 2ab + b^2) $$ 5. Sum terms: $$ a^2 + 2ab + b^2 + a^2 - b^2 + a^2 - 2ab + b^2 = 3a^2 + b^2 $$ 6. Final factorization: $$ 2b(3a^2 + b^2) $$ --- ### 9. а) $x^2 - 5x + 6$ 2. Find factors of 6 that sum to -5: -2 and -3 3. Factor: $$ (x - 2)(x - 3) $$ --- ### 9. г) $a^2 - 7ab + 10b^2$ 2. Find factors of 10 that sum to -7: -5 and -2 3. Factor: $$ (a - 5b)(a - 2b) $$ --- ### 9. ж) $a^2 - 3ab - 10b^2$ 2. Find factors of -10 that sum to -3: 2 and -5 3. Factor: $$ (a + 2b)(a - 5b) $$ --- ### 9. к) $2x^2 + 14x + 24$ 2. Factor out 2: $$ 2(x^2 + 7x + 12) $$ 3. Factor quadratic: $$ 2(x + 3)(x + 4) $$ --- ### 10. а) $a^8 + a^4 + 1$ 2. Recognize this as a quadratic in $a^4$: $$ (a^4)^2 + a^4 + 1 $$ 3. Try to factor as: $$ (a^4 + x)(a^4 + y) $$ No simple factorization over reals; leave as is. --- ### 10. б) $a^4 + a^2b^2 + b^4$ 2. Recognize as sum of squares and product: $$ (a^2 + b^2)^2 - a^2b^2 = (a^2 + b^2 - ab)(a^2 + b^2 + ab) $$ --- ### 10. в) $a^3 - 3a + 2$ 2. Try to find roots: Try $a=1$: $1 - 3 + 2 = 0$ root found. 3. Divide polynomial by $(a - 1)$: $$ a^3 - 3a + 2 = (a - 1)(a^2 + a - 2) $$ 4. Factor quadratic: $$ (a - 1)(a + 2)(a - 1) = (a - 1)^2(a + 2) $$ --- **Final answers:** - 7.а) $(x - 1)(x + 1)^2(x^2 - x + 1)$ - 7.в) $(a - 2)(a^2 + 8a + 4)$ - 7.д) $(a + 1)^2(a^2 - a + 1)$ - 7.ж) $(a + b)^2(a - b)$ - 7.б) $(m^2 + 1)(m - 1)(m^2 + m + 1)$ - 7.г) $(p + 2)^3$ - 7.е) $(x + 1)(x - 1)(x^2 + x + 1)$ - 7.з) $(x - y)^2(x + y)$ - 8.а) $(m - n)(m + n)(m^2 + n^2)$ - 8.в) $(x + 1)^2(x^2 - x + 1)$ - 8.д) $2b(3a^2 + b^2)$ - 9.а) $(x - 2)(x - 3)$ - 9.г) $(a - 5b)(a - 2b)$ - 9.ж) $(a + 2b)(a - 5b)$ - 9.к) $2(x + 3)(x + 4)$ - 10.а) No simple factorization over reals - 10.б) $(a^2 + b^2 - ab)(a^2 + b^2 + ab)$ - 10.в) $(a - 1)^2(a + 2)$