Question

1) 2а( c-d )^3+3d(c-d)^2 2)8a^3b^5+16a^4b^4-32a^5b^3 3) 25x^2-(x+y)^2 4)ax^2+bx^2-bx-ax+cx^2-cx 5)a^4+a^3+a+1

Answer 1

1-(c-d)^2x(2ac-2ad+3d)

2-

3-(4x-y)x(6x+y)

4-

5-(a+1)^2x(a^2-a+1)

Answer 2

Let's simplify each of the given expressions:

1. 2a(c - d)^3 + 3d(c - d)^2:

Expanding the terms using the binomial theorem: 2a(c^3 - 3c^2d + 3cd^2 - d^3) + 3d(c^2 - 2cd + d^2)

Simplifying by distributing the coefficients: 2ac^3 - 6ac^2d + 6acd^2 - 2ad^3 + 3dc^2 - 6d^2c + 3d^3

Rearranging the terms: 2ac^3 + 3dc^2 - 6ac^2d - 6d^2c + 6acd^2 + 3d^3 - 2ad^3

2. 8a^3b^5 + 16a^4b^4 - 32a^5b^3:

We can factor out the common term a^3b^3: 8a^3b^3(b^2) + 16a^3b^3(a) - 32a^3b^3(a^2)

Simplifying: 8a^3b^3(b^2 + 2a - 4a^2)

3. 25x^2 - (x + y)^2:

Expanding the square: 25x^2 - (x^2 + 2xy + y^2)

Distributing the negative sign: 25x^2 - x^2 - 2xy - y^2

Combining like terms: 24x^2 - 2xy - y^2

4. ax^2 + bx^2 - bx - ax + cx^2 - cx:

Rearranging the terms: (ax^2 + cx^2) + (bx^2 - bx) + (-ax - cx)

Combining like terms within each group: (a + c)x^2 + (b - b - a - c)x

Simplifying further: (a + c)x^2 - (a + c)x

Factoring out the common term (a + c): (a + c)(x^2 - x)

5. a^4 + a^3 + a + 1:

There are no like terms to combine, so this expression is already simplified.

Therefore, the simplified forms of the given expressions are:

1. 2ac^3 + 3dc^2 - 6ac^2d - 6d^2c + 6acd^2 + 3d^3 - 2ad^3

2. 8a^3b^3(b^2 + 2a - 4a^2)

3. 24x^2 - 2xy - y^2

4. (a + c)(x^2 - x)

5. a^4 + a^3 + a + 1