Question

1. subtract: $\begin{array}{r}5x^2 - 3\\ - (2x + 4)\\ hlineend{array}$
2. subtract: $\begin{array}{r}6x + 2\\ - (x^2 - 4x + 1)\\ hlineend{array}$
3. subtract: $\begin{array}{r}4a^3 + 2a\\ - (a^2 - 3a + 1)\\ hlineend{array}$
4. subtract: $\begin{array}{r}k^3 - 1\\ - (k^2 + 2k + 3)\\ hlineend{array}$
5. subtract: $\begin{array}{r}3 - y + 2y^2\\ - (y^2 + 5)\\ hlineend{array}$
6. subtract: $\begin{array}{r}m^2 - 5m\\ - (2m^3 - m + 4)\\ hlineend{array}$
7. subtract: $\begin{array}{r}n^4 - 2n^2\\ - (n^3 + 3n^2 - 4)\\ hlineend{array}$
8. subtract: $\begin{array}{r}2x^3 + 4x\\ - (3x^4 - x^2 + 2)\\ hlineend{array}$

Explanation:

Let's solve problem 25 as an example (we can solve others similarly by following the same steps of distributing the negative sign and combining like terms):

Step1: Distribute the negative sign

We have to subtract \((2x + 4)\) from \(5x^{2}-3\). So we rewrite the subtraction as addition of the opposite: \(5x^{2}-3-2x - 4\)

Step2: Combine like terms

The constant terms \(- 3\) and \(-4\) can be combined. So \(5x^{2}-2x+(-3 - 4)=5x^{2}-2x-7\)

Step1: Distribute the negative sign

We need to subtract \((a^{2}-3a + 1)\) from \(4a^{3}+2a\). So we get \(4a^{3}+2a-a^{2}+3a - 1\)

Step2: Combine like terms

The terms with \(a\) are \(2a\) and \(3a\). Combining them: \(4a^{3}-a^{2}+(2a + 3a)-1=4a^{3}-a^{2}+5a - 1\)

Step1: Distribute the negative sign

Subtract \((y^{2}+5)\) from \(3 - y+2y^{2}\). We have \(3 - y+2y^{2}-y^{2}-5\)

Step2: Combine like terms

Combine \(2y^{2}\) and \(-y^{2}\), and \(3\) and \(-5\): \( (2y^{2}-y^{2})-y+(3 - 5)=y^{2}-y - 2\)

Answer:

\(5x^{2}-2x - 7\)

If you want solutions for other problems (like 26, 27, etc.), the general steps are:

1. Distribute the negative sign to each term in the polynomial being subtracted.
2. Combine like terms (terms with the same variable and exponent) by adding or subtracting their coefficients.

For example, let's solve problem 26: