Lcm Polynomials

1. **Problem Statement:** Find the L.C.M. of the polynomial expressions given in questions 4 and 5. 2. **Formula and Rules:** - The L.C.M. (Least Common Multiple) of polynomials is the polynomial of the lowest degree that is divisible by each of the given polynomials. - To find the L.C.M., factorize each polynomial completely. - Take the highest power of each factor appearing in any polynomial. - Multiply these factors to get the L.C.M. --- ### Question 4: L.C.M. of polynomial expressions **a)** $ax^2 + ax$ and $a^2x^2 + a^2x$ - Factorize: - $ax^2 + ax = ax(x + 1)$ - $a^2x^2 + a^2x = a^2x(x + 1)$ - Highest powers: - $a^2$, $x$, $(x + 1)$ - L.C.M. = $a^2 x (x + 1)$ **b)** $2x^2 + 4x$ and $x^3 + 2x^2$ - Factorize: - $2x^2 + 4x = 2x(x + 2)$ - $x^3 + 2x^2 = x^2(x + 2)$ - Highest powers: - $2$, $x^2$, $(x + 2)$ - L.C.M. = $2 x^2 (x + 2)$ **c)** $3a^2b + 6ab^2$ and $2a^3 + 4a^2b$ - Factorize: - $3a^2b + 6ab^2 = 3ab(a + 2b)$ - $2a^3 + 4a^2b = 2a^2(a + 2b)$ - Highest powers: - $3$, $a^2$, $b$, $(a + 2b)$ - L.C.M. = $6 a^2 b (a + 2b)$ **d)** $a^2x + abx$ and $abx^2 + b^2x^2$ - Factorize: - $a^2x + abx = ax(a + b)$ - $abx^2 + b^2x^2 = b x^2 (a + b)$ - Highest powers: - $a$, $b$, $x^2$, $(a + b)$ - L.C.M. = $a b x^2 (a + b)$ **e)** $3x^2 + 6x$ and $2x^3 + 4x^2$ - Factorize: - $3x^2 + 6x = 3x(x + 2)$ - $2x^3 + 4x^2 = 2x^2(x + 2)$ - Highest powers: - $6$, $x^2$, $(x + 2)$ - L.C.M. = $6 x^2 (x + 2)$ **f)** $2a + 4$ and $a^2 - 4$ - Factorize: - $2a + 4 = 2(a + 2)$ - $a^2 - 4 = (a - 2)(a + 2)$ - Highest powers: - $2$, $(a + 2)$, $(a - 2)$ - L.C.M. = $2 (a + 2)(a - 2)$ **g)** $3a^2 + 3a$ and $6a^2 - 6$ - Factorize: - $3a^2 + 3a = 3a(a + 1)$ - $6a^2 - 6 = 6(a^2 - 1) = 6(a - 1)(a + 1)$ - Highest powers: - $6$, $a$, $(a + 1)$, $(a - 1)$ - L.C.M. = $6 a (a + 1)(a - 1)$ **h)** $x^2 y - 5 x y$ and $x^2 - 25$ - Factorize: - $x^2 y - 5 x y = x y (x - 5)$ - $x^2 - 25 = (x - 5)(x + 5)$ - Highest powers: - $x y$, $(x - 5)$, $(x + 5)$ - L.C.M. = $x y (x - 5)(x + 5)$ **i)** $a^4 x^2 - a^2 x^4$ and $7 a^2 x - 7 a x^2$ - Factorize: - $a^4 x^2 - a^2 x^4 = a^2 x^2 (a^2 - x^2) = a^2 x^2 (a - x)(a + x)$ - $7 a^2 x - 7 a x^2 = 7 a x (a - x)$ - Highest powers: - $7 a^2 x^2$, $(a - x)$, $(a + x)$ - L.C.M. = $7 a^2 x^2 (a - x)(a + x)$ **j)** $x^2 - x y$ and $x^3 y - x y^3$ - Factorize: - $x^2 - x y = x (x - y)$ - $x^3 y - x y^3 = x y (x^2 - y^2) = x y (x - y)(x + y)$ - Highest powers: - $x y$, $(x - y)$, $(x + y)$ - L.C.M. = $x y (x - y)(x + y)$ **k)** $4 x^2 - 2 x$ and $8 x^3 - 2 x$ - Factorize: - $4 x^2 - 2 x = 2 x (2 x - 1)$ - $8 x^3 - 2 x = 2 x (4 x^2 - 1) = 2 x (2 x - 1)(2 x + 1)$ - Highest powers: - $2 x$, $(2 x - 1)$, $(2 x + 1)$ - L.C.M. = $2 x (2 x - 1)(2 x + 1)$ **l)** $x^2 - 4$ and $x^2 + 5 x + 6$ - Factorize: - $x^2 - 4 = (x - 2)(x + 2)$ - $x^2 + 5 x + 6 = (x + 2)(x + 3)$ - Highest powers: - $(x - 2)$, $(x + 2)$, $(x + 3)$ - L.C.M. = $(x - 2)(x + 2)(x + 3)$ **m)** $x^2 + x - 6$ and $x^2 - 9$ - Factorize: - $x^2 + x - 6 = (x + 3)(x - 2)$ - $x^2 - 9 = (x - 3)(x + 3)$ - Highest powers: - $(x + 3)$, $(x - 2)$, $(x - 3)$ - L.C.M. = $(x + 3)(x - 2)(x - 3)$ **n)** $2 x^3 - 50 x$ and $2 x^2 + 7 x - 15$ - Factorize: - $2 x^3 - 50 x = 2 x (x^2 - 25) = 2 x (x - 5)(x + 5)$ - $2 x^2 + 7 x - 15 = (2 x - 3)(x + 5)$ - Highest powers: - $2 x$, $(x - 5)$, $(x + 5)$, $(2 x - 3)$ - L.C.M. = $2 x (x - 5)(x + 5)(2 x - 3)$ **o)** $4 a^3 - 9 a$ and $2 a^2 + 3 a - 9$ - Factorize: - $4 a^3 - 9 a = a (4 a^2 - 9) = a (2 a - 3)(2 a + 3)$ - $2 a^2 + 3 a - 9 = (2 a - 3)(a + 3)$ - Highest powers: - $a$, $(2 a - 3)$, $(2 a + 3)$, $(a + 3)$ - L.C.M. = $a (2 a - 3)(2 a + 3)(a + 3)$ **p)** $x^2 + 8 x + 15$ and $x^2 + 7 x + 12$ - Factorize: - $x^2 + 8 x + 15 = (x + 3)(x + 5)$ - $x^2 + 7 x + 12 = (x + 3)(x + 4)$ - Highest powers: - $(x + 3)$, $(x + 4)$, $(x + 5)$ - L.C.M. = $(x + 3)(x + 4)(x + 5)$ **q)** $a^2 - 9 a + 20$ and $a^2 - 2 a - 15$ - Factorize: - $a^2 - 9 a + 20 = (a - 4)(a - 5)$ - $a^2 - 2 a - 15 = (a - 5)(a + 3)$ - Highest powers: - $(a - 4)$, $(a - 5)$, $(a + 3)$ - L.C.M. = $(a - 4)(a - 5)(a + 3)$ **r)** $a^2 + 5 a - 14$ and $a^2 - 8 a + 12$ - Factorize: - $a^2 + 5 a - 14 = (a + 7)(a - 2)$ - $a^2 - 8 a + 12 = (a - 6)(a - 2)$ - Highest powers: - $(a + 7)$, $(a - 2)$, $(a - 6)$ - L.C.M. = $(a + 7)(a - 2)(a - 6)$ **s)** $2 a^2 + 5 a + 2$ and $2 a^2 - 3 a - 2$ - Factorize: - $2 a^2 + 5 a + 2 = (2 a + 1)(a + 2)$ - $2 a^2 - 3 a - 2 = (2 a + 1)(a - 2)$ - Highest powers: - $(2 a + 1)$, $(a + 2)$, $(a - 2)$ - L.C.M. = $(2 a + 1)(a + 2)(a - 2)$ **t)** $3 x^2 + 8 x - 16$ and $3 x^2 - 16 x + 16$ - Factorize: - $3 x^2 + 8 x - 16 = (3 x - 4)(x + 4)$ - $3 x^2 - 16 x + 16 = (3 x - 4)(x - 4)$ - Highest powers: - $(3 x - 4)$, $(x + 4)$, $(x - 4)$ - L.C.M. = $(3 x - 4)(x + 4)(x - 4)$ **u)** $4 x^2 - x - 3$ and $3 x^2 - 2 x - 1$ - Factorize: - $4 x^2 - x - 3 = (4 x + 3)(x - 1)$ - $3 x^2 - 2 x - 1 = (3 x + 1)(x - 1)$ - Highest powers: - $(4 x + 3)$, $(3 x + 1)$, $(x - 1)$ - L.C.M. = $(4 x + 3)(3 x + 1)(x - 1)$ **v)** $2 x^2 + 3 x - 9$ and $4 x^2 - 12 x + 9$ - Factorize: - $2 x^2 + 3 x - 9 = (2 x - 3)(x + 3)$ - $4 x^2 - 12 x + 9 = (2 x - 3)^2$ - Highest powers: - $(2 x - 3)^2$, $(x + 3)$ - L.C.M. = $(2 x - 3)^2 (x + 3)$ --- ### Question 5: L.C.M. of polynomial expressions **a)** $m^2 + 2 m$, $m^2 - 4$, $m^2 + 3 m + 2$ - Factorize: - $m^2 + 2 m = m(m + 2)$ - $m^2 - 4 = (m - 2)(m + 2)$ - $m^2 + 3 m + 2 = (m + 1)(m + 2)$ - Highest powers: - $m$, $(m + 2)$, $(m - 2)$, $(m + 1)$ - L.C.M. = $m (m + 2)(m - 2)(m + 1)$ **b)** $\\sqrt{4 x - 20}$, $x^2 - 25$, $x^2 - 3 x - 10$ - Note: $\\sqrt{4 x - 20} = \\sqrt{4(x - 5)} = 2 \\sqrt{x - 5}$ - Factorize: - $x^2 - 25 = (x - 5)(x + 5)$ - $x^2 - 3 x - 10 = (x - 5)(x + 2)$ - L.C.M. must include $2 \\sqrt{x - 5}$ and the other factors: - L.C.M. = $2 \\sqrt{x - 5} (x + 5)(x + 2)$ **c)** $(a + b)^2$, $a^2 - b^2$, $2 a^2 + a b - b^2$ - Factorize: - $(a + b)^2$ is already factored - $a^2 - b^2 = (a - b)(a + b)$ - $2 a^2 + a b - b^2 = (2 a - b)(a + b)$ - Highest powers: - $(a + b)^2$, $(a - b)$, $(2 a - b)$ - L.C.M. = $(a + b)^2 (a - b)(2 a - b)$ **d)** $x^2 + 4 x + 4$, $x^2 - 4$, $x^2 + 8 x + 12$ - Factorize: - $x^2 + 4 x + 4 = (x + 2)^2$ - $x^2 - 4 = (x - 2)(x + 2)$ - $x^2 + 8 x + 12 = (x + 6)(x + 2)$ - Highest powers: - $(x + 2)^2$, $(x - 2)$, $(x + 6)$ - L.C.M. = $(x + 2)^2 (x - 2)(x + 6)$ **e)** $x^2 - 6 x + 9$, $x^2 - 9$, $x^2 - 10 x + 21$ - Factorize: - $x^2 - 6 x + 9 = (x - 3)^2$ - $x^2 - 9 = (x - 3)(x + 3)$ - $x^2 - 10 x + 21 = (x - 7)(x - 3)$ - Highest powers: - $(x - 3)^2$, $(x + 3)$, $(x - 7)$ - L.C.M. = $(x - 3)^2 (x + 3)(x - 7)$ **f)** $x^2 + 6 x + 8$, $x^2 + x - 12$, $x^2 + 5 x + 6$ - Factorize: - $x^2 + 6 x + 8 = (x + 4)(x + 2)$ - $x^2 + x - 12 = (x + 4)(x - 3)$ - $x^2 + 5 x + 6 = (x + 3)(x + 2)$ - Highest powers: - $(x + 4)$, $(x + 2)$, $(x - 3)$, $(x + 3)$ - L.C.M. = $(x + 4)(x + 2)(x - 3)(x + 3)$ **g)** $a^2 + 4 a - 5$, $a^2 + 11 a + 30$, $a^2 - 5 a - 6$ - Factorize: - $a^2 + 4 a - 5 = (a + 5)(a - 1)$ - $a^2 + 11 a + 30 = (a + 5)(a + 6)$ - $a^2 - 5 a - 6 = (a - 6)(a + 1)$ - Highest powers: - $(a + 5)$, $(a - 1)$, $(a + 6)$, $(a + 1)$ - L.C.M. = $(a + 5)(a - 1)(a + 6)(a + 1)$ **h)** $2 a^2 + 5 a + 2$, $3 a^2 - 7 a + 2$, $6 a^2 + 5 a + 1$ - Factorize: - $2 a^2 + 5 a + 2 = (2 a + 1)(a + 2)$ - $3 a^2 - 7 a + 2 = (3 a - 1)(a - 2)$ - $6 a^2 + 5 a + 1 = (3 a + 1)(2 a + 1)$ - Highest powers: - $(2 a + 1)$, $(a + 2)$, $(3 a - 1)$, $(a - 2)$, $(3 a + 1)$ - L.C.M. = $(2 a + 1)(a + 2)(3 a - 1)(a - 2)(3 a + 1)$ --- **Final answers:** - L.C.M. for each polynomial pair/triple is the product of the highest powers of all unique factors from the factorization.