**Problem Set 3.2: Addition and Subtraction of Polynomials** 1. Add $(3m - 5k - h) + (-6m + 4k - 5h)$ 1. Combine like terms: $$3m + (-6m) = -3m$$ $$-5k + 4k = -k$$ $$-h + (-5h) = -6h$$ Answer: $$-3m - k - 6h$$ 2. Compute $(9a + 8b + 7c) + (4a + 3b + 2c) - (-11a - 11b - 8c)$ 1. Add first two polynomials: $$9a + 4a = 13a$$ $$8b + 3b = 11b$$ $$7c + 2c = 9c$$ So sum is $$13a + 11b + 9c$$ 2. Subtract third polynomial (note minus sign): $$13a - (-11a) = 13a + 11a = 24a$$ $$11b - (-11b) = 11b + 11b = 22b$$ $$9c - (-8c) = 9c + 8c = 17c$$ Answer: $$24a + 22b + 17c$$ 3. Compute $(p - q) - (p - q)$ 1. Distribute minus sign: $$(p - q) - p + q$$ 2. Combine like terms: $$p - p = 0$$ $$-q + q = 0$$ Answer: $$0$$ 4. Compute $(4z^5 - 6z^3 + z - 8) - (2z^5 + 7z^4 - 3z^2 + 4z + 1)$ 1. Distribute minus: $$4z^5 - 6z^3 + z - 8 - 2z^5 - 7z^4 + 3z^2 - 4z - 1$$ 2. Combine like terms: $$4z^5 - 2z^5 = 2z^5$$ $$-7z^4$$ $$-6z^3$$ $$3z^2$$ $$z - 4z = -3z$$ $$-8 - 1 = -9$$ Answer: $$2z^5 - 7z^4 - 6z^3 + 3z^2 - 3z - 9$$ 5. Simplify $$9x + 7y - 3y - 4x - 5x - 4y$$ 1. Combine like terms: $$9x - 4x - 5x = 0$$ $$7y - 3y - 4y = 0$$ Answer: $$0$$ 6. Compute $(4x^3 - 7x^2 + 2x - 4) + (3x^3 + 8x^2 + 3x - 7)$ 1. Add corresponding terms: $$4x^3 + 3x^3 = 7x^3$$ $$-7x^2 + 8x^2 = x^2$$ $$2x + 3x = 5x$$ $$-4 - 7 = -11$$ Answer: $$7x^3 + x^2 + 5x - 11$$ 7. Compute $(32x^3 + 53x^2y - 73xy^2) - (5x^2y - 71xy^2 + 26y^3) + (x^3 - y^3)$ 1. Expand: $$32x^3 + 53x^2y - 73xy^2 - 5x^2y + 71xy^2 - 26y^3 + x^3 - y^3$$ 2. Combine like terms: $$32x^3 + x^3 = 33x^3$$ $$53x^2y - 5x^2y = 48x^2y$$ $$-73xy^2 + 71xy^2 = -2xy^2$$ $$-26y^3 - y^3 = -27y^3$$ Answer: $$33x^3 + 48x^2y - 2xy^2 - 27y^3$$ 8. Compute $(x^3 - 11x^2 + 22x - 14) + (5x^3 + 6x^2 + 3x - 4)$ 1. Add each term: $$x^3 + 5x^3 = 6x^3$$ $$-11x^2 + 6x^2 = -5x^2$$ $$22x + 3x = 25x$$ $$-14 - 4 = -18$$ Answer: $$6x^3 - 5x^2 + 25x - 18$$ 9. Compute $(x^3 - 64x^2 - 99x + 18) - (5x^3 + 9x^2 - 14) - (4x^3 - 4x^2 - 75x - 6)$ 1. Distribute minuses: $$x^3 - 64x^2 - 99x + 18 - 5x^3 - 9x^2 + 14 - 4x^3 + 4x^2 + 75x + 6$$ 2. Combine like terms: $$x^3 - 5x^3 - 4x^3 = -8x^3$$ $$-64x^2 - 9x^2 + 4x^2 = -69x^2$$ $$-99x + 75x = -24x$$ $$18 + 14 + 6 = 38$$ Answer: $$-8x^3 - 69x^2 - 24x + 38$$ 10. Subtract $(4ax - 5a^2x - 8ax^2)$ from $(3ax + 7a^2x + 6ax^2) + (-8x - 9a^2x + 2ax^2)$ 1. Sum the two polynomials inside parentheses: $$3ax + 7a^2x + 6ax^2 - 8x - 9a^2x + 2ax^2 =$$ $$3ax + (7a^2x - 9a^2x) + (6ax^2 + 2ax^2) - 8x =$$ $$3ax - 2a^2x + 8ax^2 - 8x$$ 2. Subtract $(4ax - 5a^2x - 8ax^2)$: $$3ax - 2a^2x + 8ax^2 - 8x - 4ax + 5a^2x + 8ax^2 =$$ $$ (3ax - 4ax) + (-2a^2x + 5a^2x) + (8ax^2 + 8ax^2) - 8x =$$ $$-ax + 3a^2x + 16ax^2 - 8x$$ Answer: $$-ax + 3a^2x + 16ax^2 - 8x$$ **Problem Set 3.3: Multiplication of Polynomials** 1. Multiply $(12x^2y^3)(-5x^2y^4)$ 1. Multiply coefficients: $$12 imes (-5) = -60$$ 2. Add exponents for like bases: $$x^{2+2} = x^4$$ $$y^{3+4} = y^7$$ Answer: $$-60x^4y^7$$ 2. Multiply $6xy^3z(2x^2y - 3yz^2 + 4xz)$ Multiply each term: $$6xy^3z imes 2x^2y = 12x^{1+2}y^{3+1}z = 12x^3y^4z$$ $$6xy^3z imes (-3yz^2) = -18xy^{3+1}z^{1+2} = -18xy^4z^3$$ $$6xy^3z imes 4xz = 24x^{1+1}y^3z^{1+1} = 24x^2y^3z^2$$ Answer: $$12x^3y^4z - 18xy^4z^3 + 24x^2y^3z^2$$ 3. Multiply $$-8x^4y^3z^2(7x^2y^3z - 5y^3z^2 + 11xyz - 9)$$ Multiply each term: $$-8x^4y^3z^2 imes 7x^2y^3z = -56x^{4+2}y^{3+3}z^{2+1} = -56x^6y^6z^3$$ $$-8x^4y^3z^2 imes (-5y^3z^2) = 40x^4y^{3+3}z^{2+2} = 40x^4y^6z^4$$ $$-8x^4y^3z^2 imes 11xyz = -88x^{4+1}y^{3+1}z^{2+1} = -88x^5y^4z^3$$ $$-8x^4y^3z^2 imes (-9) = 72x^4y^3z^2$$ Answer: $$-56x^6y^6z^3 + 40x^4y^6z^4 - 88x^5y^4z^3 + 72x^4y^3z^2$$ 4. Multiply $(5m - 3n)(4m + 7n)$ Use distributive property: $$5m imes 4m = 20m^2$$ $$5m imes 7n = 35mn$$ $$-3n imes 4m = -12mn$$ $$-3n imes 7n = -21n^2$$ Combine like terms: $$35mn - 12mn = 23mn$$ Answer: $$20m^2 + 23mn - 21n^2$$ 5. Multiply $(3m^2 + 2n)(3m^2 - 2n)$ This is a difference of squares pattern: $$(3m^2)^2 - (2n)^2 = 9m^4 - 4n^2$$ Answer: $$9m^4 - 4n^2$$ 6. Compute $(12x^3y^2 - z^2)^2$ Square the binomial: $$(12x^3y^2)^2 - 2 imes 12x^3y^2 imes z^2 + (z^2)^2 =$$ $$144x^6y^4 - 24x^3y^2z^2 + z^4$$ Answer: $$144x^6y^4 - 24x^3y^2z^2 + z^4$$ 7. Compute $(2b^2 - 3a^2)^3$ Use binomial cube formula: $$a^3 - 3a^2b + 3ab^2 - b^3$$ For $$a = 2b^2, b = 3a^2$$ Calculate each term: $$a^3 = (2b^2)^3 = 8b^6$$ $$3a^2b = 3 imes (2b^2)^2 imes 3a^2 = 3 imes 4b^4 imes 3a^2 = 36a^2b^4$$ $$3ab^2 = 3 imes 2b^2 imes (3a^2)^2 = 3 imes 2b^2 imes 9a^4 = 54a^4b^2$$ $$b^3 = (3a^2)^3 = 27a^6$$ Use alternating signs: $$8b^6 - 36a^2b^4 + 54a^4b^2 - 27a^6$$ Answer: $$8b^6 - 36a^2b^4 + 54a^4b^2 - 27a^6$$ 8. Compute $(5x^2 - 2x - 3)^2$ Square trinomial: $$(5x^2)^2 + 2 imes 5x^2 imes (-2x) + 2 imes 5x^2 imes (-3) + (-2x)^2 + 2 imes (-2x) imes (-3) + (-3)^2$$ Calculate terms: $$= 25x^4 - 20x^3 - 30x^2 + 4x^2 + 12x + 9$$ Combine like terms: $$25x^4 - 20x^3 - 26x^2 + 12x + 9$$ Answer: $$25x^4 - 20x^3 - 26x^2 + 12x + 9$$ 9. Compute $(5x - 3y + 4z)^2$ Square trinomial: $$(5x)^2 + (-3y)^2 + (4z)^2 + 2 imes 5x imes (-3y) + 2 imes 5x imes 4z + 2 imes (-3y) imes 4z$$ Calculate terms: $$25x^2 + 9y^2 + 16z^2 - 30xy + 40xz - 24yz$$ Answer: $$25x^2 + 9y^2 + 16z^2 - 30xy + 40xz - 24yz$$ 10. Multiply $(3x^4 + 2x^3 - 7x + 1)(-3x^4 - 7x^2 + 5x - 2)$ Distribute each term and combine: - $3x^4 imes -3x^4 = -9x^8$ - $3x^4 imes -7x^2 = -21x^6$ - $3x^4 imes 5x = 15x^5$ - $3x^4 imes -2 = -6x^4$ - $2x^3 imes -3x^4 = -6x^7$ - $2x^3 imes -7x^2 = -14x^5$ - $2x^3 imes 5x = 10x^4$ - $2x^3 imes -2 = -4x^3$ - $-7x imes -3x^4 = 21x^5$ - $-7x imes -7x^2 = 49x^3$ - $-7x imes 5x = -35x^2$ - $-7x imes -2 = 14x$ - $1 imes -3x^4 = -3x^4$ - $1 imes -7x^2 = -7x^2$ - $1 imes 5x = 5x$ - $1 imes -2 = -2$ Combine like terms: - $-9x^8$ - $-6x^7$ - $-21x^6$ - $15x^5 - 14x^5 + 21x^5 = 22x^5$ - $-6x^4 + 10x^4 - 3x^4 = 1x^4$ - $-4x^3 + 49x^3 = 45x^3$ - $-35x^2 - 7x^2 = -42x^2$ - $14x + 5x = 19x$ - $-2$ Answer: $$-9x^8 - 6x^7 - 21x^6 + 22x^5 + x^4 + 45x^3 - 42x^2 + 19x - 2$$