Answer:
Multiple answers
Step-by-step explanation:
For the 1st question: you're going to want to use the swing method. Here's how you do it.
1) you want to get the m^2 alone so you multiply 7 by 8 to get m^2 alone.
You should get m^2+18m+56. Then factor that by finding two numbers that will add up to 18 and also when you multiply them you get 56. Those two numbers are 9 and 7. So you set it up like (m+9)(m+7). But you're not done there. You have to divide 9 and 7 by 7 because you multiplied by 7 in the very beginning. (m+9/7)(m+7/7) which you will get (7m+9)(m+1)!
2) You do the same thing in the first question. You multiply the 7 by 6 to get the x^2 alone. You should get x^2+43x+42. Find the two numbers that will add up to 43 but also if you multiply those two numbers you get 42. You should then get 42 and 1 as your factors. Then set it up like this: (x+42)(x+1) but remember you have to divide by 7. (x+42/7)(x+1/7) which should lead you to get (x+6)(7x+1). If you're wondering how I got 7x+1, it's because I just moved the 7 from the denominator and put it in front of x.
3) You once again multiply 7 by 60 to get n^2 alone. 7 times 60 = 420. Find those two factors which should add up and multiply to get 52. Those two factors should be -42 and -10. Set it up like this: (x-42)(x-10) and then you have divide by 7. (x-42/7)(x-10/7) which equals to (x-6)(7x+1).
4) This once is simpler since the a^2 is already alone. To solve this, you just need to find the two factors that will add up to -11 but also if you multiply them you'll get 30. The factors should be -5 and -6. (x-6)(x-5) and that's your answer!
5) Just like all of the previous questions, you need to multiply 3 by 30 to get the x^2 alone. 3 times 30 = 90! So now you need to find two factors that add up to 23 but also multiplying those two numbers will get you 90. Those two factors should be: 5 and 18. (x+5)(x+18) and divide those numbers by 3. (x+5/3)(x+18/3) which equals to (3x+5)(x+6) and that's your answer.
6) For the last question, you need to multiply 5 by -32 to get the a^2 alone. 5 times -32 = -160. The two factors will be (n-4)(n+40) and divide them by 5 the original number you multiplied with in the beginning. (n-4/5)(n+40/5) which equals to (5n-4)(n+8) and that's your answer!
Answer:
1) the factors are (7m + 4)(m + 2)
2) the factors are (7x + 1)(x + 6)
3) the factors are (7n - 10)(n - 6)
4) the factors are (a - 6)(a - 5)
5) the factors are (3x + 5)(x + 6)
6) the factors are (5n - 4)(n + 8)
Step-by-step explanation:
* To factor a trinomial in the form ax² ± bx ± c:
- Look at the c term first.
# If the c term is a positive number, then the factors of c will both be positive or both be negative. In other words, r and s will have the same sign and find two integers, r and s, whose product is c , h and k, whose product is a and the sum of c and a is b.
# If the c term is a negative number, then one factor of c will be positive, and one factor of c will be negative. Either r or s will be negative, but not both.and find two integers, r and s, whose product is c , h and k, whose product is a and the difference of c and a is b.
- Look at the b term second.
# If the c term is positive and the b term is positive, then both r and s are positive.
Ex: 2x² + 8x + 6 = (2x + 2)(x + 3)
# If the c term is positive and the b term is negative, then both r and s are negative.
Ex: 2x² - 8x + 6 = (2x - 2)(x - 3)
# If the c term is negative and the b term is positive, then the factor that is positive will have the greater absolute value. That is, if |sh| > |rk|, then s is positive and r is negative.
Ex: 2x² + 4x - 6 = (2x - 2)(x + 3)
# If the c term is negative and the b term is negative, then the factor that is negative will have the greater absolute value. That is, if |sh| > |rk|, then s is negative and r is positive.
Ex: 2x² - 4x - 6 = (2x + 2)(x - 3)
* Now lets factorize the problems
1) 7m² + 18m + 8
- The sign of the two brackets is +ve
∵ 7m² = 7m × m
∵ 8 = 2 × 4
∵ 7m × 2 = 14m
∵ m × 4 = 4m
∵ 14m + 4m = 18m
∴ the factors are (7m + 4)(m + 2) ⇒ (h with s and r with k)
2) 7x² + 43x + 6
- The sign of the two brackets is +ve
∵ 7x² = 7x × x
∵ 6 = 1 × 6
∵ 7x × 6 = 42x
∵ x × 1 = x
∵ 42x + x = 43x
∴ the factors are (7x + 1)(x + 6) ⇒ (h with s and r with k)
3) 7n² - 52n + 60
- The sign of the two brackets is -ve
∵ 7n² = 7n × n
∵ 60 = 10 × 6
∵ 7n × 6 = 42n
∵ n × 10 = 10n
∵ 42n + 10n = 52n
∴ the factors are (7n - 10)(n - 6) ⇒ (h with s and r with k)
4) a² - 11a + 30
- The sign of the two brackets is -ve
∵ a² = a × a
∵ 30 = 6 × 5
∵ a × 6 = 6a
∵ a × 5 = 5a
∵ 6a + 5a = 11a
∴ the factors are (a - 6)(a - 5)
5) 3x² + 23x + 30
- The sign of the two brackets is +ve
∵ 3x² = 3x × x
∵ 30 = 5 × 6
∵ 3x × 6 = 18x
∵ x × 5 = 5x
∵ 18x + 5x = 23x
∴ the factors are (3x + 5)(x + 6) ⇒ (h with s and r with k)
6) 5n² + 36n - 32
- The sign of the two brackets one is -ve and one is +ve
∵ 5n² = 5n × n
∵ 32 = 4 × 8
∵ 5n × 8 = 40n
∵ n × 4 = 4n
∵ 40n - 4n = 36n
∴ the factors are (5n - 4)(n + 8) ⇒ (h with s and r with k)
