Question

Simply the following polynomials.
13. (x+4)^2
16. (x+1) (x-1)
17. (y-8) (y+8)
18. (2w+1)(2w-1)
19. (3a+4b) (3a-4b)
21. (m+3)(m^2+4m+7)
22. (4x-1)(x^2-7x+1)
25.5x+2+x+x+7+3x-1

Answer 1

Let's simplify each of the given polynomials one by one:

**13. \((x+4)^2\)**
Using the formula \( (a+b)^2 = a^2 + 2ab + b^2 \):
\[
(x+4)^2 = x^2 + 2 \cdot x \cdot 4 + 4^2 = x^2 + 8x + 16
\]

**16. \((x+1)(x-1)\)**
Using the difference of squares formula \( a^2 - b^2 \):
\[
(x+1)(x-1) = x^2 - 1
\]

**17. \((y-8)(y+8)\)**
Using the difference of squares formula:
\[
(y-8)(y+8) = y^2 - 64
\]

**18. \((2w+1)(2w-1)\)**
Using the difference of squares formula:
\[
(2w+1)(2w-1) = (2w)^2 - (1)^2 = 4w^2 - 1
\]

**19. \((3a+4b)(3a-4b)\)**
Using the difference of squares formula:
\[
(3a+4b)(3a-4b) = (3a)^2 - (4b)^2 = 9a^2 - 16b^2
\]

**21. \((m+3)(m^2+4m+7)\)**
Distributing \(m\) and \(3\):
\[
m(m^2 + 4m + 7) + 3(m^2 + 4m + 7) = m^3 + 4m^2 + 7m + 3m^2 + 12m + 21
\]
Combining like terms:
\[
m^3 + (4m^2 + 3m^2) + (7m + 12m) + 21 = m^3 + 7m^2 + 19m + 21
\]

**22. \((4x-1)(x^2-7x+1)\)**
Distributing \(4x\) and \(-1\):
\[
4x(x^2 - 7x + 1) - 1(x^2 - 7x + 1) = 4x^3 - 28x^2 + 4x - x^2 + 7x - 1
\]
Combining like terms:
\[
4x^3 + (-28x^2 - x^2) + (4x + 7x) - 1 = 4x^3 - 29x^2 + 11x - 1
\]

**25. \(5x + 2 + x + x + 7 + 3x - 1\)**
Combining like terms:
\[
(5x + x + x + 3x) + (2 + 7 - 1) = (5x + 1x + 1x + 3x) + (2 + 7 - 1) = 10x + 8
\]

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### Final Simplified Results:
13. \(x^2 + 8x + 16\)
16. \(x^2 - 1\)
17. \(y^2 - 64\)
18. \(4w^2 - 1\)
19. \(9a^2 - 16b^2\)
21. \(m^3 + 7m^2 + 19m + 21\)
22. \(4x^3 - 29x^2 + 11x - 1\)
25. \(10x + 8\)