8xy+4y^2 and -9p^3+37p^2-18p and 6x^2+7x-49 and 15n^2-27n-6

Question

Please show work 

Answer 1

I assume you are required to factor these equations.  You need to state what you are trying to accomplish...

 

1)   8xy + 4y²

      Look for common factors ... things that can be divided out of both terms.

      4 is a common factor and y is a common factor so factor out 4y

      8xy + 4y² = 4y(2x + y)

 

2)  -9p³ + 37p² -18p

     Since 37 is a prime number and only factors to (1) (37) we can't factor out a constant so look for

     any variables in common.  In this case each term can be divided by p so...

     -9p³ + 37p² - 18p = p(-9p² + 37p -18)

      To see if we can factor further we look at the product of the coefficient of the squared term and the constant

      (-9)(-18) = 162    so we look for any factors of 162 that will add to the coefficient of the middle p term 37

      I am unable to come up with any so I don't believe this can be factored further

    

 

3)  6x² + 7x -49

     Since we can't divide out a common factor from all terms look at the product of the first coefficient 6 from the 6x² and the end constant (-49).   ...   6(-49) = -294

 

What we want are factors of -294 that will add to make 7 because the middle term is 7x

Those factors are (21)(-14) because (21)(-14) = -294 and 21-14=7

 

What we do is we use those two factors to split up the middle term 7x into 21x - 14x and then factor the result.

We can set it up as either 6x² +21x -14x -49   or    6x² -14x + 21x -49

Looking at the two possibilities I see the most obvious factors will be to group the 6x² with 21x and the -14x with -49

because those two combinations have the most obvious common factors so lets try...

6x² + 21x - 14x -49

From the first two terms factor out 3x  and from the 2nd two terms factor out -7 giving...

3x(2x+7) - 7(2x+7)

We can use the identity that a(c) + b(c) = (a+b)(c) to get

3x(2x+7) - 7(2x+7) = (3x-7)(2x+7)

 

4)  15n² -27n - 6

     We can see that 3 is a common factor so let's factor that out first giving

     3(5n² - 9n -2)

     Now let's follow the same process as on the previous problem

     Since (5)(-2) = -10 we need factors of -10 that add to -9   .... that would be (-10)(1)

     So lets change the -9n to -10n +n

    3(5n² - 9n -2) = 3(5n² - 10n + n - 2)

    Again let's factor the first 2 terms in the parentheses and the 2nd two terms...

    We can factor 5n from the 1st two terms and 1 from the 2nd two terms so...

    3(5n² - 10n + n - 2) = 3[5n(n-2) + 1(n-2)]

     15n² -27n - 6= 3(5n+1)(n-2)