Question

add and subtract polynomials
2/6/2026 test
name: breysi
simplify each expression to its simplest form.

1. $(4x - 3) - (2x + 1)$
2. $(3x^3 + 5x - 4) + (5x^2 - 14x + 2)$
3. $(-6v^4 + 5v^3 - v) - (-5v^4 + 5v^4 - v)$
4. $(8 - 5k^2) - (7k^2 + 5)$
5. $2x^2 + 5x) + (7 + 5x^2 - x)$

Explanation:

Problem 1: $(4x - 3) - (2x + 1)$

Step 1: Distribute the negative sign

To simplify the expression, we first distribute the negative sign to the terms inside the second parentheses. This gives us:
$$(4x - 3) - 2x - 1$$

Step 2: Combine like terms

Next, we combine the like terms. The like terms for the \(x\)-terms are \(4x\) and \(-2x\), and the constant terms are \(-3\) and \(-1\). Combining the \(x\)-terms: \(4x - 2x = 2x\). Combining the constant terms: \(-3 - 1 = -4\). So the simplified expression is:
$$2x - 4$$

Step 1: Remove the parentheses

Since we are adding the two polynomials, we can simply remove the parentheses:
$$3x^3 + 5x - 4 + 5x^2 - 14x + 2$$

Step 2: Combine like terms

Now we combine the like terms. The \(x^3\)-term is \(3x^3\), the \(x^2\)-term is \(5x^2\), the \(x\)-terms are \(5x\) and \(-14x\), and the constant terms are \(-4\) and \(2\). Combining the \(x\)-terms: \(5x - 14x = -9x\). Combining the constant terms: \(-4 + 2 = -2\). So the simplified expression is:
$$3x^3 + 5x^2 - 9x - 2$$

Step 1: Simplify the second parentheses

First, we simplify the terms inside the second parentheses. Combining the \(v^4\)-terms: \(-5v^4 + 5v^4 = 0\). So the second parentheses simplifies to \(0 - v = -v\). Now our expression becomes:
$$(-6v^4 + 5v^3 - v) - (-v)$$

Step 2: Distribute the negative sign

Distribute the negative sign to the term inside the second parentheses:
$$-6v^4 + 5v^3 - v + v$$

Step 3: Combine like terms

The \(v\)-terms \(-v\) and \(v\) cancel each other out. So we are left with:
$$-6v^4 + 5v^3$$

Answer:

$2x - 4$

Problem 2: $(3x^3 + 5x - 4) + (5x^2 - 14x + 2)$