1. Problem 27: Find the difference $(3c^3 - c + 11) - (c^2 + 2c + 8)$. Step 1: Distribute the minus sign to the second polynomial: $$3c^3 - c + 11 - c^2 - 2c - 8$$ Step 2: Combine like terms: - For $c^3$: $3c^3$ - For $c^2$: $-c^2$ - For $c$: $-c - 2c = -3c$ - Constants: $11 - 8 = 3$ Final answer: $$3c^3 - c^2 - 3c + 3$$ 2. Problem 28: Find the sum $(z^2 + z) + (z^2 - 11)$. Step 1: Combine like terms: - For $z^2$: $z^2 + z^2 = 2z^2$ - For $z$: $z$ - Constants: $-11$ Final answer: $$2z^2 + z - 11$$ 3. Problem 29: Find the difference $(2x - 2y + 1) - (3y + 4x)$. Step 1: Distribute the minus sign: $$2x - 2y + 1 - 3y - 4x$$ Step 2: Combine like terms: - For $x$: $2x - 4x = -2x$ - For $y$: $-2y - 3y = -5y$ - Constants: $1$ Final answer: $$-2x - 5y + 1$$ 4. Problem 30: Find the sum $(4a - 5b^2 + 3) + (6 - 2a + 3b^2)$. Step 1: Combine like terms: - For $a$: $4a - 2a = 2a$ - For $b^2$: $-5b^2 + 3b^2 = -2b^2$ - Constants: $3 + 6 = 9$ Final answer: $$2a - 2b^2 + 9$$ 5. Problem 31: Find the sum $(x^2y - 3x^2 + y) + (3y - 2x^2y)$. Step 1: Combine like terms: - For $x^2y$: $x^2y - 2x^2y = -x^2y$ - For $x^2$: $-3x^2$ - For $y$: $y + 3y = 4y$ Final answer: $$-x^2y - 3x^2 + 4y$$ 6. Problem 32: Find the sum $(-8xy + 3x^2 - 5y) + (4x^2 - 2y + 6xy)$. Step 1: Combine like terms: - For $xy$: $-8xy + 6xy = -2xy$ - For $x^2$: $3x^2 + 4x^2 = 7x^2$ - For $y$: $-5y - 2y = -7y$ Final answer: $$-2xy + 7x^2 - 7y$$ 7. Problem 33: Find the difference $(5n - 2p^2 + 2np) - (4p^2 + 4n)$. Step 1: Distribute the minus sign: $$5n - 2p^2 + 2np - 4p^2 - 4n$$ Step 2: Combine like terms: - For $n$: $5n - 4n = n$ - For $p^2$: $-2p^2 - 4p^2 = -6p^2$ - For $np$: $2np$ Final answer: $$n - 6p^2 + 2np$$ 8. Problem 34: Find the difference $(4rxt - 8r^2x + x^2) - (6rx^2 + 5rxt - 2x^2)$. Step 1: Distribute the minus sign: $$4rxt - 8r^2x + x^2 - 6rx^2 - 5rxt + 2x^2$$ Step 2: Combine like terms: - For $rxt$: $4rxt - 5rxt = -rxt$ - For $r^2x$: $-8r^2x$ - For $x^2$: $x^2 + 2x^2 = 3x^2$ - For $rx^2$: $-6rx^2$ Final answer: $$-rxt - 8r^2x + 3x^2 - 6rx^2$$