MTH
030 -- Elementary Algebra -- Exercise Solutions
Section:
5.2
30)
Factor x^{2} + 7x + 10 Solution: x^{2} + 7x + 10 = (x + 2)(x + 5) justify by multiplying |
Find two numbers whose product is 10 and whose sum is 7. They will be the constant terms in the linear factors. So the constant terms will be 2 and 5. |
32)
Factor n^{2} - 7n + 10 |
Find two numbers whose product is 10 and whose sum is - 7. They will be the constant terms in the linear factors. Because 10 is positive, the two numbers must have the same sign and since - 7 is negative that sign is minus. So the constant terms will be - 2 and - 5. |
34)
Factor b^{2} + 6b - 7 |
Find a positive number and a negative number whose product is -7 and whose sum is 6. They will be the constant terms in the linear factors. Because - 7 is negative, the two numbers must have different signs and since +6 is positive the larger of the two is positive. So the constant terms will be 7 and - 1. |
36)
Factor t^{2} - 5t - 50 |
Find a positive number and a negative number whose product is - 50 and whose sum is - 5. They will be the constant terms in the linear factors. Because - 50 is negative, the two numbers must have different signs and since - 5 is negative the larger of the two is negative. So the constant terms will be -10 and 5. |
38)
Factor r^{2} - 9r - 12 |
Find a positive number and a negative number whose product is - 12and whose sum is - 9. They will be the constant terms in the linear factors. Because - 12 is negative, the two numbers must have different signs and since - 9 is negative the larger of the two is negative. No pair of divisors of 12 can fit these conditions so the trinomial is prime. |
40)
Factor v^{2} + 9v + 15 |
Find two numbers whose product is 15 and whose sum is 9. They will be the constant terms in the linear factors. No pair of divisors of 15 can fit these conditions so the trinomial is prime. |
42)
Factor y^{2} + 8y + 12 justify by multiplying |
Find two numbers whose product is 12 and whose sum is 8. They will be the constant terms in the linear factors. So the constant terms will be 2 and 6. |
44)
Factor m^{2} + 3m - 10 |
Find a positive number and a negative number whose product is - 10 and whose sum is 3. They will be the constant terms in the linear factors. Because - 10 is negative, the two numbers must have different signs and since + 3 is positive the larger of the two is positive. So the constant terms will be 5 and - 2. |
46)
Factor u^{2} + u - 42 |
Find a positive number and a negative number whose product is - 42 and whose sum is 1. They will be the constant terms in the linear factors. Because - 42 is negative, the two numbers must have different signs and since + 1 is positive the larger of the two is positive. So the constant terms will be 7 and - 6. |
48)
Factor a^{2} + 10ab + 9b^{2} |
Find two numbers whose product is 9 and whose sum is 10. They will be the coefficients of b in the linear factors. So the second terms will be 9b and b. |
50)
Factor m^{2} - mn - 12n^{2} |
Find a positive number and a negative number whose product is - 12 and whose sum is -1. They will be the coefficients of n in the linear factors. Because - 12 is negative, the two numbers must have different signs and since - 1 is negative the larger of the two is negative. So the second terms will be - 4n and 3n. |
52)
Factor p^{2} + pq - 6q^{2} |
Find a positive number and a negative number whose product is - 6 and whose sum is 1. They will be the coefficients of q in the linear factors. Because - 6 is negative, the two numbers must have different signs and since + 1 is positive the larger of the two is positive. So the second terms will be 3q and - 2q. |
54)
Factor m^{2} + 3mn - 20n^{2} |
Find a positive number and a negative number whose product is - 20 and whose sum is 3. They will be the coefficients of q in the linear factors. No pair of divisors of 15 can fit these conditions so the trinomial is prime. |
56)
Factor -x^{2} + 9x - 20 |
58)
Factor - t^{2} - t + 30 |
60)
Factor - r^{2} + 14r - 45 |
62)
Factor - a^{2} - 6ab - 5b^{2} Solution: - a^{2} - 6ab - 5b^{2} = (-1)[a^{2} + 6ab + 5b^{2}] = (-1)[(a +5b)(a + b)] = (-1)(a +5b)(a + b) = - (a +5b)(a + b) justify by multiplying |
64)
Factor - x^{2} - 10xy + 11y^{2} |
66)
Factor y^{2} + 5 + 6y Solution: y^{2} + 5 + 6y = y^{2} + 6y + 5 = (y + 5)(y + 1) justify by multiplying |
68)
Factor x^{2} - 13 - 12x |
70)
Factor u^{2} - 3 + 2u Solution: u^{2} - 3 + 2u = u^{2} + 2u - 3 = (u + 3)(u - 1) justify by multiplying |
72)
Factor a^{2} + 5b^{2} + 6ab |
74)
Factor -13yz + y^{2} - 14z^{2} Solution: -13yz + y^{2} - 14z^{2} = y^{2}- 13yz - 14z^{2} = (y - 14z)(y + z) justify by multiplying |
76)
Factor 3y^{2} - 21y + 18 |
78)
Factor - 2b^{2} + 20b - 18 Solution: - 2b^{2} + 20b - 18 = (-2)[b^{2} - 10b + 9] = (-2)[(b - 9)(b - 1)] = (-2)(b - 9)(b - 1) justify by multiplying |
80)
Factor 5m^{2} + 45m - 50 |
82)
Factor 48xy + 6xy^{2} + 96x |
84)
Factor 3x^{2}y^{3} + 3x^{3}y^{2}
- 6xy^{4} |