1. Add the polynomials. **a)** $(3x^2 - x + 2) + (4x^2 + 3x - 1)$ Combine like terms: $$3x^2 + 4x^2 = 7x^2$$ $$-x + 3x = 2x$$ $$2 - 1 = 1$$ Answer: $$7x^2 + 2x + 1$$ **b)** $(2t^2 + 5t - 7) + (3t^2 - 4t + 6)$ Combine like terms: $$2t^2 + 3t^2 = 5t^2$$ $$5t - 4t = t$$ $$-7 + 6 = -1$$ Answer: $$5t^2 + t - 1$$ **c)** $(7m^2 - mn - 8n^2) + (6m^2 + 9mn + 11n^2)$ Combine like terms: $$7m^2 + 6m^2 = 13m^2$$ $$-mn + 9mn = 8mn$$ $$-8n^2 + 11n^2 = 3n^2$$ Answer: $$13m^2 + 8mn + 3n^2$$ **d)** $(-4y^2 + 2xy - 6x^2) + (5y^2 - 6xy + 7y^2)$ Combine like terms: $$-4y^2 + 5y^2 + 7y^2 = 8y^2$$ $$2xy - 6xy = -4xy$$ $$-6x^2$$ (no like term) Answer: $$8y^2 - 4xy - 6x^2$$ **e)** $(3xy - 2x + 7) + (6xy + 5x - 3)$ Combine like terms: $$3xy + 6xy = 9xy$$ $$-2x + 5x = 3x$$ $$7 - 3 = 4$$ Answer: $$9xy + 3x + 4$$ **f)** $(5x + 3y - 8xy) + (6xy + 2x - 5y)$ Combine like terms: $$5x + 2x = 7x$$ $$3y - 5y = -2y$$ $$-8xy + 6xy = -2xy$$ Answer: $$7x - 2y - 2xy$$ 2. Subtract the polynomials. **a)** $(3x^2 - 7x + 3) - (x^2 + 5x - 2)$ Distribute minus: $$3x^2 - 7x + 3 - x^2 - 5x + 2$$ Combine like terms: $$3x^2 - x^2 = 2x^2$$ $$-7x - 5x = -12x$$ $$3 + 2 = 5$$ Answer: $$2x^2 - 12x + 5$$ **b)** $(5s^3 + 8s - 12) - (6s^3 - 5 + 4)$ Distribute minus: $$5s^3 + 8s - 12 - 6s^3 + 5 - 4$$ Combine like terms: $$5s^3 - 6s^3 = -s^3$$ $$8s$$ (no like term) $$-12 + 5 - 4 = -11$$ Answer: $$-s^3 + 8s - 11$$ **c)** $(9x^2 - 4xy - y^2) - (6y^2 + 3xy + 10x^2)$ Distribute minus: $$9x^2 - 4xy - y^2 - 6y^2 - 3xy - 10x^2$$ Combine like terms: $$9x^2 - 10x^2 = -x^2$$ $$-4xy - 3xy = -7xy$$ $$-y^2 - 6y^2 = -7y^2$$ Answer: $$-x^2 - 7xy - 7y^2$$ **d)** $(-r^2 + 4rs + s) - (6r^2 - rs + 11s^2)$ Distribute minus: $$-r^2 + 4rs + s - 6r^2 + rs - 11s^2$$ Combine like terms: $$-r^2 - 6r^2 = -7r^2$$ $$4rs + rs = 5rs$$ $$s - 11s^2$$ (no like terms) Answer: $$-7r^2 + 5rs + s - 11s^2$$ **e)** $(3x + 4y - 5z) - (x - y - z)$ Distribute minus: $$3x + 4y - 5z - x + y + z$$ Combine like terms: $$3x - x = 2x$$ $$4y + y = 5y$$ $$-5z + z = -4z$$ Answer: $$2x + 5y - 4z$$ **f)** $(5m - 3n) - (2m - 7n + 4)$ Distribute minus: $$5m - 3n - 2m + 7n - 4$$ Combine like terms: $$5m - 2m = 3m$$ $$-3n + 7n = 4n$$ $$-4$$ (constant) Answer: $$3m + 4n - 4$$ 3. Add the sum of $3x^2 - 6x + 5$ and $-3x^2 + 6$ to $-x^2 - x - 1$. First sum: $$3x^2 - 6x + 5 + (-3x^2 + 6) = (3x^2 - 3x^2) + (-6x) + (5 + 6) = 0 - 6x + 11 = -6x + 11$$ Add to $-x^2 - x - 1$: $$-x^2 - x - 1 + (-6x + 11) = -x^2 + (-x - 6x) + (-1 + 11) = -x^2 - 7x + 10$$ Answer: $$-x^2 - 7x + 10$$ 4. Add $4x + 2y - 7$ to the sum of $-2x + 3y - 2$ and $3x + y - 4$. Sum inside parentheses: $$-2x + 3y - 2 + 3x + y - 4 = (-2x + 3x) + (3y + y) + (-2 - 4) = x + 4y - 6$$ Add $4x + 2y - 7$: $$4x + 2y - 7 + x + 4y - 6 = (4x + x) + (2y + 4y) + (-7 - 6) = 5x + 6y - 13$$ Answer: $$5x + 6y - 13$$ 5. Subtract $3t^2 + 4t - 7$ from the sum of $2t^2 - 5t + 3$ and $4t^2 + 2t + 3$. Sum: $$2t^2 - 5t + 3 + 4t^2 + 2t + 3 = (2t^2 + 4t^2) + (-5t + 2t) + (3 + 3) = 6t^2 - 3t + 6$$ Subtract $3t^2 + 4t - 7$: $$6t^2 - 3t + 6 - (3t^2 + 4t - 7) = 6t^2 - 3t + 6 - 3t^2 - 4t + 7 = (6t^2 - 3t^2) + (-3t - 4t) + (6 + 7) = 3t^2 - 7t + 13$$ Answer: $$3t^2 - 7t + 13$$ 6. Subtract the sum of $m^2 + 2m - 3$ and $4m^2 - m + 2$ from $3m^2 + 4m - 1$. Sum: $$m^2 + 2m - 3 + 4m^2 - m + 2 = (m^2 + 4m^2) + (2m - m) + (-3 + 2) = 5m^2 + m - 1$$ Subtract from $3m^2 + 4m - 1$: $$3m^2 + 4m - 1 - (5m^2 + m - 1) = 3m^2 + 4m - 1 - 5m^2 - m + 1 = (3m^2 - 5m^2) + (4m - m) + (-1 + 1) = -2m^2 + 3m + 0 = -2m^2 + 3m$$ Answer: $$-2m^2 + 3m$$ 7. The perimeter of a triangle is $5x - 2y + 3z$. Two sides are $3y + z$ and $4x - y + z$. Find the third side. Let the third side be $S$. Perimeter formula: $$S + (3y + z) + (4x - y + z) = 5x - 2y + 3z$$ Combine known sides: $$(3y + z) + (4x - y + z) = 4x + (3y - y) + (z + z) = 4x + 2y + 2z$$ So: $$S + 4x + 2y + 2z = 5x - 2y + 3z$$ Solve for $S$: $$S = 5x - 2y + 3z - 4x - 2y - 2z = (5x - 4x) + (-2y - 2y) + (3z - 2z) = x - 4y + z$$ Answer: $$x - 4y + z$$