How can I rewrite the middle term as the sum of two terms and then factor completley?

Question

12w^2 + 19w + 4

&

4z^3 - 18z^2 - 10z

Answer 1

You are breaking the middle to find the products that will sum to the middle

(dw+e)(fw+g)
= (d*f)w^2 + (d*g)w + (e*f)w + (e*g)
D*f=12
e*g=4
d*g + e*f =19

also, look at the signs, since all of the coefficients are positive, then all of the scalar variables are also positive.

so

12w^2 + 16w + 3w + 4
or
(4w+1)(3w+4)

The second is still really a quadratic:
z(4z^2-18z-10)
simplify
2z(2z^2-9z-5)

2z(2z +z-10z-5)
2z(2z+1)(z-5)

j

Answer 2

~ element through Grouping ~ ax² + bx + c (a million) Get the grouping volume (ac). (2) locate the element pair of the grouping volume whose sum is an identical as (b). (3) Rewrite the expression using those elements. (4) element through grouping. 10x² + x - 2 (a million) 10*-2 = -20 --- Grouping volume. (2) 5 and -4 --- because (5) + (-4) = a million --- element pair whose sum = b (3) 10x² + 5x - 4x - 2 --- Rewrite the equation using those elements. (4) 5x(2x + a million) - 2(2x + a million) --- Take 5x out of 10x² and 5x, and take -2 out of -4x² and -2. (5x - 2)(2x + a million)