Question

Given the trinomial, what is the value of the coefficient B in the factored form?

2x2 + 4xy − 48y2 = 2(x + By)(x − 4y)

Answer 1

`\huge\boxed{B=6}`

Explanation:

They are two way to solution.

METHOD 1:

Factor the polynomial on the left side of the equation:

`2x^2+4xy-48y^2=2(x^2+2xy-24y^2)=2(x^2+6xy-4xy-24y^2)\\\\=2\bigg(x(x+6y)-4y(x+6y)\bigg)=2(x+6y)(x-4y)`

Therefore:

`2x^2+4xy-48y^2=2(x+By)(x-4y)\\\Downarrow\\2(x+6y)(x-4y)=2(x+By)(x-4y)\to\boxed{\bold{B=6}}`

METHOD 2:

Multiply everything on the right side of the equation using the distributive property and FOIL:

`2(x+By)(x-4y)=\bigg((2)(x)+(2)(By)\bigg)(x-4y)\\\\=(2x+2By)(x-4y)=(2x)(x)+(2x)(-4y)+(2By)(x)+(2By)(-4y)\\\\=2x^2-8xy+2Bxy-8By^2=2x^2+(2B-8)xy-8By^2`

Compare polynomials:

`2x^2+4xy-48y^2=2x^2+(2B-8)xy-8By^2`

From here we have two equations:

`2B-8=4\ \text{and}\ -8B=-48`

`1)\\2B-8=4` add 8 to both sides

`2B=12` divide both sides by 2

`B=6`

`2)\\-8B=-48` divide both sides by (-8)

`B=6`

The results are the same. Therefore B = 6.

Answer 2

B = 6

Explanation:

2x^2 + 4xy − 48y^2

Factor out 2

2(x^2 + 2xy − 24y^2)

What 2 numbers multiply to -24 and add to 2

-4 *6 = -24

-4+6 = 2

2 ( x+6y)( x-4y)