Question

x^4 +16x^2 will be perfect
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Answer 1

Step-by-step explanation:

STEP

1

:

Equation at the end of step 1

((x4) - 24x2) + 64 = 0

STEP

2

:

Trying to factor by splitting the middle term

2.1 Factoring x4-16x2+64

The first term is, x4 its coefficient is 1 .

The middle term is, -16x2 its coefficient is -16 .

The last term, "the constant", is +64

Step-1 : Multiply the coefficient of the first term by the constant 1 • 64 = 64

Step-2 : Find two factors of 64 whose sum equals the coefficient of the middle term, which is -16 .

-64 + -1 = -65

-32 + -2 = -34

-16 + -4 = -20

-8 + -8 = -16 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -8 and -8

x4 - 8x2 - 8x2 - 64

Step-4 : Add up the first 2 terms, pulling out like factors :

x2 • (x2-8)

Add up the last 2 terms, pulling out common factors :

8 • (x2-8)

Step-5 : Add up the four terms of step 4 :

(x2-8) • (x2-8)

Which is the desired factorization

Trying to factor as a Difference of Squares:

2.2 Factoring: x2-8

Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)

Proof : (A+B) • (A-B) =

A2 - AB + BA - B2 =

A2 - AB + AB - B2 =

A2 - B2

Note : AB = BA is the commutative property of multiplication.