Answer:

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. - AB + AB equals zero and is therefore eliminated from the expression.

Given : 16m² - 25 n²

Check :

= 16 is the square of 4

= 25 is the square of 5

= m2 is the square of m1

Final result : (4m + 5n) • (4m - 5n)

Given: 4x² - 20x + 25

Factoring 4x²-20x+25

The first term is, 4x² its coefficient is 4 .

The middle term is, -20x its coefficient is -20 .

The last term, "the constant", is +25

Step-1 : Multiply the coefficient of the first term by the constant 4 • 25 = 100

Step-2 : Find two factors of 100 whose sum equals the coefficient of the middle term, which is -20 .

-100 + -1 = -101

-50 + -2 = -52

-25 + -4 = -29

-20 + -5 = -25

-10 + -10 = -20 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -10 and -10

4x² - 10x - 10x - 25

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

2x • (2x-5)

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

5 • (2x-5)

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

(2x-5) • (2x-5)

Which is the desired factorization

Multiplying Exponential Expressions:

2.2 Multiply (2x-5) by (2x-5)

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is (2x-5) and the exponents are :

1 , as (2x-5) is the same number as (2x-5)¹

and 1 , as (2x-5) is the same number as (2x-5)¹

The product is therefore, (2x-5)(1+1) = (2x-5)²

Final result : (2x - 5)²

Given: 3x² - x - 2

Factoring 3x2-x-2

The first term is, 3x2 its coefficient is 3 .

The middle term is, -x its coefficient is -1 .

The last term, "the constant", is -2

Step-1 : Multiply the coefficient of the first term by the constant 3 • -2 = -6

Step-2 : Find two factors of -6 whose sum equals the coefficient of the middle term, which is -1 .

-6 + 1 = -5

-3 + 2 = -1 That's it

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

3x2 - 3x + 2x - 2

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

3x • (x-1)

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

2 • (x-1)

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

(3x+2) • (x-1)

Which is the desired factorization

Final result : (x - 1) • (3x + 2)

Given: 6x²y6z² + 18x³y²z - 15x²yz⁴

STEP 1

:

Equation at the end of step 1

((((6•(x2))•(y6))•(z2))+(((18•(x3))•(y2))•z))-(((3•5x2)•y)•z4)

STEP 2

:

((((6•(x2))•(y6))•(z2))+(((2•32x3)•y2)•z))-(3•5x2yz4)

STEP 3

:

((((2•3x2) • y6) • z2) + (2•32x3y2z)) - (3•5x2yz4)

STEP 4

:

Pulling out like terms

5.1 Pull out like factors :

18x3y2z + 6x2y6z2 - 15x2yz4 = 3x2yz • (6xy + 2y5z - 5z3)

Trying to factor a multi variable polynomial :

5.2 Factoring 6xy + 2y5z - 5z3

Try to factor this multi-variable trinomial using trial and error

Final result : 3x2yz • (6xy + 2y5z - 5z3)

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