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3 years ago

15 points for 1 question??

Mathematics

1 answer:

mestny [16]3 years ago

5 0

Answer:a7b2-ab2

Final result :

ab2•(a+1)•(a2-a+1)•(a-1)•(a2+a+1)

Step by step solution :

Step 1 :

Step 2 :

Pulling out like terms :

2.1 Pull out like factors :

a7b2 - ab2 = ab2 • (a6 - 1)

Trying to factor as a Difference of Squares :

2.2 Factoring: a6 - 1

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.

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

Check : 1 is the square of 1

Check : a6 is the square of a3

Factorization is : (a3 + 1) • (a3 - 1)

Trying to factor as a Sum of Cubes :

2.3 Factoring: a3 + 1

Theory : A sum of two perfect cubes, a3 + b3 can be factored into :

(a+b) • (a2-ab+b2)

Proof : (a+b) • (a2-ab+b2) =

a3-a2b+ab2+ba2-b2a+b3 =

a3+(a2b-ba2)+(ab2-b2a)+b3=

a3+0+0+b3=

a3+b3

Check : 1 is the cube of 1

Check : a3 is the cube of a1

Factorization is :

(a + 1) • (a2 - a + 1)

Trying to factor by splitting the middle term

2.4 Factoring a2 - a + 1

The first term is, a2 its coefficient is 1 .

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

The last term, "the constant", is +1

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

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

-1 + -1 = -2

1 + 1 = 2

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Trying to factor as a Difference of Cubes:

2.5 Factoring: a3-1

Theory : A difference of two perfect cubes, a3 - b3 can be factored into

(a-b) • (a2 +ab +b2)

Proof : (a-b)•(a2+ab+b2) =

a3+a2b+ab2-ba2-b2a-b3 =

a3+(a2b-ba2)+(ab2-b2a)-b3 =

a3+0+0+b3 =