How do I factor this polynomial: 2x2−5xy−y2?

Question

trnovitom06 · Accepted Answer

How would you factor it if the y werent

tainiaadjouctlw · Accepted Answer

To explain how you get from your polynomial to a more traditional quadratic equation: pull out a factor of

y^{2}

to get

y^{2} \cdot (2 \frac{x^{2}}{y^{2}} - 5 \frac{x y}{y^{2}} - \frac{y^{2}}{y^{2}})

, or equivalently

y^{2} \cdot (2 \frac{x^{2}}{y^{2}} - 5 \frac{x}{y} - 1)

. Now, setting

t = \frac{x}{y}

, the latter is obviously

y^{2} \cdot (2 t^{2} - 5 t - 1)

. The quadratic can be broken up into two factors by solving the quadratic equation as other answers have mentioned:

2 t^{2} - 5 t - 1 = (a t + b) (c t + d)

. Then plugging this expression back in, we get the original function as

y^{2} \cdot (a t + b) (c t + d), or (y \cdot (a t + b)) (y \cdot (c t + d))

;
plugging in x/y for t and multiplying through by the factors of y then gives the factorization as

(a x + b y) (c x + d y)

RizerMix · Accepted Answer

Consider 2x2−5xy−y2=0. Solving this quadratic equation for x gives x=5y+33y4,5y−33y4. Let 5+334=α and 5−334=β. Thus, factorization of given polynomial is 2(x−αy)(x−βy).

How do I factor this polynomial: 2x^{2}-5xy-y^{2}?

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