Factor completely

Question

2a⁴-9a²+4 

Answer 1

Since 'a' always occurs as an even power of 2, let c=a^2, so we can transform the nastier looking 4th degree polynomial into a 2nd degree polynomial, which is much easier to tackle (attempt to factor into two binomials).

So now we have the simpler equation 2c^2 - 9c + 4.

The term 4 is factored by either 1 and 4 or 2 and 2, either both positive or both negative.  I'll guess 1 and 4 and notice that since I need the middle term to be negative, I need to choose -1 and -4.  The first term has a coefficient of 2, and to get the middle coefficient to be -9, I'll try the following and check using the standard FOIL method:

 

(2c - 1)(c - 4)   <=== check using FOIL: 2c^2  + (2c)x(-4) + (-1)x(c)  + 4,

                                                             FIRST  +  OUTER   +   INNER    LAST

                                                             

It works!  We got the 2nd degree polynomial for 'c' factored into two binomials.

Finally, let's substitute back c=a^2 to obtain

(2a^2-1)(a^2-4)

 

This is nice, but gee, we know (a^2-b^2)=(a-b)(a+b), [here b^=4, so b=2], so we can further factor the above as

(2a^2-1)(a-2)(a+2) 

 

How cool!  We only needed to identify that 'a' always occurs as a power of 2 (a^4, a^2, a^0) to make the trivial substitution and solve the problem using the standard algebra I(or II) procedure for factoring.  We ended up with one 2nd degree polynomial and two binomials.

 

 

Now you can see how easy it would be to solve 2a^8 - 9a^4 + 4 by playing the same substitution game.

(let c = a^4 so you have 2c^2 - 9c + 4.  Then have again (2c - 1)(c - 4) and back substitute to get

(2a^4 - 1)(a^4 - 4).

 

play the game again on (a^4 - 4).  Let d=a^2 so you have (d^2-4)=(d - 2)(d + 2) and back substitute again to get (a^4 - 4) = (a^2 - 2)(a^2 + 2).  So you finally get 

2a^8 - 9a^4 + 4 = (2a^4 - 1)(a^2 - 2)(a^2 + 2). 

 

Here we "keep it real" and factored an octic polynomial into a quartic and two quadradic polynomials.

Answer 2

factoring is done by inspection; there is no method nor standard procedure.

in this case 2a⁴ - 9a² + 4, the first step gives us (2a² - 1)(a² - 4)

 

[the key here is the coefficient 9, because the only way to obtain that value, given the other coefficients, is to multiply 2 x 4, and then add 1]

[also, the minus sign followed by a plus sign tells us that the two coefficients used to produce +4 are both minus]

 

but, (a² - 4) can be factored further, to (a + 2)(a - 2)

 

thus, the factors are (2a² - 1)(a + 2)(a - 2)

 

we can check our work by multiplying the factors: (a + 2)(a - 2) yields a² - 4, which, when multiplied by (2a² - 1), yields the original expression