Question: Simplify the following polynomials. Write
1. $3m + 4m - (-6m) + (-8m)$
2. $2x - 13y + 9x + 5y$
3. $4x^2 + 6x - 5 + 3x + (-2x^2) - 6$
4. $a^2 - 12a + 9 - (5a - 3a^2 + 2)$
5. $y + 3y - (-6y) + (-4y)$
6. $5m - 3m^2 + 2m - m^2 + 9m^2$
7. $(x^5y)(xy^3)$
8. $-2(m^3n^3)(-4mn^2)$
1. **Simplify** $3m + 4m - (-6m) + (-8m)$
Step 1: Remove parentheses and change signs where needed:
$3m + 4m + 6m - 8m$
Step 2: Combine like terms (all terms have $m$):
$3m + 4m = 7m$
$7m + 6m = 13m$
$13m - 8m = 5m$
**Answer:** $5m$
2. **Simplify** $2x - 13y + 9x + 5y$
Step 1: Group like terms:
$(2x + 9x) + (-13y + 5y)$
Step 2: Combine:
$11x - 8y$
**Answer:** $11x - 8y$
3. **Simplify** $4x^2 + 6x - 5 + 3x + (-2x^2) - 6$
Step 1: Group like terms:
$(4x^2 - 2x^2) + (6x + 3x) + (-5 - 6)$
Step 2: Combine:
$2x^2 + 9x - 11$
**Answer:** $2x^2 + 9x - 11$
4. **Simplify** $a^2 - 12a + 9 - (5a - 3a^2 + 2)$
Step 1: Distribute the minus sign:
$a^2 - 12a + 9 - 5a + 3a^2 - 2$
Step 2: Group like terms:
$(a^2 + 3a^2) + (-12a - 5a) + (9 - 2)$
Step 3: Combine:
$4a^2 - 17a + 7$
**Answer:** $4a^2 - 17a + 7$
5. **Simplify** $y + 3y - (-6y) + (-4y)$
Step 1: Remove parentheses and change signs:
$y + 3y + 6y - 4y$
Step 2: Combine like terms:
$y + 3y = 4y$
$4y + 6y = 10y$
$10y - 4y = 6y$
**Answer:** $6y$
6. **Simplify** $5m - 3m^2 + 2m - m^2 + 9m^2$
Step 1: Group like terms:
$(5m + 2m) + (-3m^2 - m^2 + 9m^2)$
Step 2: Combine:
$7m + 5m^2$
**Answer:** $7m + 5m^2$
7. **Simplify** $(x^5y)(xy^3)$
Step 1: Multiply coefficients (both 1) and variables by adding exponents:
$x^{5+1} y^{1+3} = x^6 y^4$
**Answer:** $x^6 y^4$
8. **Simplify** $-2(m^3n^3)(-4mn^2)$
Step 1: Multiply coefficients:
$-2 imes -4 = 8$
Step 2: Multiply variables by adding exponents:
$m^{3+1} n^{3+2} = m^4 n^5$
Step 3: Combine:
$8 m^4 n^5$
**Answer:** $8 m^4 n^5$