Question

hw: simplify each expression.

1. $(8x + 3 + 5x^2) + (5x^4 - 8 - 5x^2)$
2. $(x^4 + 8x^2 - 4) + (4x + 6x^4 - 7)$
3. $(3x^3 + 3 - 2x^4) - (2 + 6x^3 - x^4)$
4. $(5x - 8x^4 + 8) + (7x + 7x^4 - 3)$
5. $(7x^2 - 7x^4 + x^3) - (2x^2 + 2x^4 - 6x^3)$
6. $(7x - 8x^2 + 3x^3) - (3x^3 + 2x^2$
7. $(x^3 - 5x + 4 - 7x^2) - (7x + 6x^3 + 4 + 6x^2)$
8. $(2x - 4x^3 - 2x^4 + 3x^2) + (5 - 6x^3 - 4x^2 + 8x)$

Explanation:

Problem 21:

Step1: Remove parentheses

\((8x + 3 + 5x^2) + (5x^4 - 8 - 5x^2)=8x + 3 + 5x^2 + 5x^4 - 8 - 5x^2\)

Step2: Combine like terms

For \(x^2\) terms: \(5x^2-5x^2 = 0\)
For constant terms: \(3 - 8=-5\)
The remaining terms: \(5x^4+8x\)
So the simplified expression is \(5x^4 + 8x-5\)

Step1: Remove parentheses

\((x^4 + 8x^2 - 4)+(4x + 6x^4 - 7)=x^4 + 8x^2 - 4+4x + 6x^4 - 7\)

Step2: Combine like terms

For \(x^4\) terms: \(x^4+6x^4 = 7x^4\)
For constant terms: \(-4-7=-11\)
The remaining terms: \(7x^4+8x^2 + 4x-11\)

Step1: Distribute the negative sign

\((3x^3 + 3 - 2x^4)-(2 + 6x^3 - x^4)=3x^3 + 3 - 2x^4-2 - 6x^3 + x^4\)

Step2: Combine like terms

For \(x^4\) terms: \(-2x^4+x^4=-x^4\)
For \(x^3\) terms: \(3x^3-6x^3=-3x^3\)
For constant terms: \(3 - 2 = 1\)
The simplified expression is \(-x^4-3x^3 + 1\)

Answer:

\(5x^4 + 8x - 5\)

Problem 22: