Factor the following polynomial.

Question

Step1: When factoring a trinomial in the form of x^2 + bx + c , you need to find the values of p and q in the factored form (x +- p) (x +-q). The product of p and q is the constant term of the trinomial. The sum or difference of p and q is the coefficient of the x-term in the trinomial.

 

Fill in the following table with all of the possible values of p and q, using the fact that their product is the constant term 56.

(table below)

| p | q | p+q   |

|    |     |          |

|    |     |          |

|    |     |          |

|    |     |          |

|    |     |          |

|    |     |          |

|    |     |          |

 

Step2: Which factors for p and q add to equal a coefficient of -15 for the x-term in the trinomial? 

 

Step3: Using the factors from Step 2, write the trinomial x^2 - 15x + 56 in factored form.

 

Step4: Follow Steps 1 through 3 to factor the trinomial x^2 + 12x +36. Show your work.

 

(By the way it was impossible to make a proper table on here and in step 1 the minus is supposed to be under the plus)

Answer 1

STEP 1

 

For Step 1, you just want to find all combinations of two numbers that multiply together to be 56. Make sure you take negatives into account. Those pairs of numbers are:

 

(p, q, p+q)

(1, 56, 57)

(-1, -56, -57)

(2, 28, 30)

(-2, -28, -30)

(4, 14, 18)

(-4, -14, -18)

(7, 8, 15)

(-7, -8, -15)

 

Use those pairs of numbers and their sums to fill in the three columns of your table.

 

 

 

STEP 2

 

Look down your p+q column to find the pair of numbers that add up to equal -15. That's (p, q) = (-7, -8).

 

 

 

STEP 3

 

Using the information from steps 1 and 2, you can factor the trinomial like this:

 

x² - 15x + 56 = (x + p)(x + q) = (x - 7)(x - 8)

 

FOIL that back out if you'd like, to make sure the expressions are equivalent.

 

 

 

STEP 4

 

Hopefully, seeing steps 1 through 3 worked out will help you with step 4. Find all the pairs of numbers that multiply together to be 36 (1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 and 6, and all their negative equivalents), then figure out which pair has a sum of 12. That means you've found your p and q, and can factor the trinomial as (x + p)(x + q).

 

 

 

 

(Yes, it's totally impossible to make a good table on here. I don't know why that is, and I wish they would fix it. For the record, yours looks great! And the ± sign is found in the symbols menu next to the "x²" button for tutors... There's probably something equivalent for students.)