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I. Simplify the following polynomial divided by monomial:
(i) 15m3 + 9m2 + 6m by 3m
(ii) 5x3 + 30x2 + 15x by 5x
(iii) 48x3 – 16x2 + 80x by 16x
(iv) -3y6 + 6y4 + y2 + 4 by 2y2
(v) 14a2b – 16ab – 20ab2 by 2ab
(vi) 14x3y3 + 21x4y2 – 49x2y4 by -7x2y2

Solution:

(i) Given that,
15m3 + 9m2 + 6m by 3m
=15m3 + 9m2 + 6m/3m
=15m3/3m + 9m2/3m + 6m/3m
=5m2 + 3m + 2
Therefore, By dividing 15m3 + 9m2 + 6m by 3m we get  5m2 + 3m + 2.
(ii) Given that,
5x3 + 30x2 + 15x by 5x
=5x3 + 30x2 + 15x/5x
=5x3/5x + 30x2/5x + 15x/5x
=x2 + 6x +3
Therefore, By dividing 5x3 + 30x2 + 15x by 5x we get  x2 + 6x +3.
(iii) Given that, 48x3 – 16x2 + 80x by 16x
=48x3 – 16x2 + 80x/16x
=48x3/16x -16x2/16x + 80x/16x
=3x2-x +5
Therefore, By dividing 48x3 – 16x2 + 80x by 16x we get 3x2-x +5.
(iv) Given that, -3y6 + 6y4 + y2 + 4 by 2y2
=-3y6 + 6y4 + y2 + 4/2y2
=-3y6/2y2 + 6y4/2y2 + y2/2y2 +4/2y2
=-3/2y4 + 3y2 + 1/2 +2/y2
Therefore, By dividing -3y6 + 6y4 + y2 + 4 by 2y2 we get -3/2y4 + 3y2 + 1/2 +2/y2.
(v) Given that, 14a2b – 16ab – 20ab2 by 2ab
=14a2b – 16ab – 20ab2/2ab
=14a2b/2ab – 16ab/2ab – 20ab2/2ab
=7a-8-10b
Therefore, By dividing 14a2b – 16ab – 20ab2 by 2ab we get 7a-8-10b.
(vi) Given that, 14x3y3 + 21x4y2 – 49x2y4 by -7x2y2
=14x3y3/-7x2y2 + 21x4y2/-7x2y2-49x2y4/-7x2y2
=-2xy-3x2+7y2
Therefore, By dividing 14x3y3 + 21x4y2 – 49x2y4 by -7x2y2 we get -2xy-3x2+7y2.

II. Solve the following by dividing the polynomial by a monomial:
(i) (x2 – 5xy) ÷ 2x
(ii) (3z3 – 6z2 + 12z) ÷ 3z
(iii) (4m6 – 3m5 + 8m4) ÷ m2
(iv) (8a7 – 6a6 + 2a4) ÷ a3
(v) (12y5 – 21y4) ÷ (-3y3)
(vi) (36a6 – 72a5) ÷ 9a5
(vii) (x4-3x3+4x2+2x) ÷x2

Solution:

(i) Given that, (x2 – 5xy) ÷ 2x
=x2/2x-5xy/2x
=1/2x-5/2y
Therefore, By dividing (x2 – 5xy) ÷ 2x we get 1/2x-5/2y.
(ii) Given that, (3z3 – 6z2 + 12z) ÷ 3z
=3z3/3z-6z2/3z +12z/3z
=z2-6z+4
Therefore, By dividing 3z3 – 6z2 + 12z by 3z we get z2-6z+4.

(iii) Given that, (4m6 – 3m5 + 8m4) ÷ m2
=4m6/m2-3m5/m2 + 8m4/m2
=4m4-3m3 + 8m2
Therefore, By dividing 4m6 – 3m5 + 8m4 with m2 we get 4m4-3m3 + 8m2.

(iv) Given that, (8a7 – 6a6 + 2a4) ÷ a3
=8a7/a3 -6a6/a3 + 2a4/a3
=8a4-6a3+2a
Therefore, By dividing 8a7 – 6a6 + 2a4 with a3 we get 8a4-6a3+2a.

(v) Given that, (12y5 – 21y4) ÷ (-3y3)
=12y5 /-3y3 + 21y4/3y3
=-4y2 +7y
Therefore, By dividing 12y5 – 21y4 with -3y3 we get -4y2 +7y.

(vi) Given that, (36a6 – 72a5) ÷ 9a5
=36a6 /9a5 – 72a5/9a5
=4a-8
Therefore, By dividing 36a6 – 72a5 with 9a5 we get 4a-8.
(vii) Given that, (x4-3x3+4x2+2x) ÷x
=x4/x-3x3/x+4x2/x+2x/x
=x3-3x2+4x+2
Therefore, By dividing x4-3x3+4x2+2x by x we get x3-3x2+4x+2.

III. Divide the following polynomial by monomial and write the answer in simplest form:
(i) 8a3 – 48a2 + 64a by 8a
(ii) 18m2n2 – 2mn2 + 6mn3 by 2mn
(iii) 8a2b – 4ab2 – 20ab by 4ab
(iv) 6x4 – 3x3 + (3/2)x2 by 3x
(v) x4 + 2x2 by x2
(vi) 5x3+ 25x2+30x by 5x

Solution:

(i) Given that, 8a3 – 48a2 + 64a by 8a
=8a3/8a-48a2/8a + 64a/8a
=a2-6a+8
Therefore, By dividing 8a3 – 48a2 + 64a by 8a we get a2-6a+8.
(ii) Given that, 18m2n2 – 2mn2 + 6mn3 by 2mn
=18m2n2/2mn-2mn2/2mn + 6mn3/2mn
=9mn-n+3n2
Therefore, By dividing 18m2n2 – 2mn2 + 6mn3 by 2mn we get 9mn-n+3n2.
(iii) Given that, 8a2b – 4ab2 – 20ab by 4ab
=8a2b/4ab-4ab2/4ab – 20ab/4ab
=2a-b-5
Therefore, By dividing 8a2b – 4ab2 – 20ab by 4ab we get 2a-b-5.
(iv) Given that, 6x4 – 3x3 + (3/2)x2 by 3x
=6x4/3x-3x3/3x + (3/2)x2/3x
=2x3 – x2 + 1/3x
Therefore, By dividing 6x4 – 3x3 + (3/2)x2 by 3x we get 2x3 – x2 + 1/3x.
(v) Given that, x4 + 2x2 by x2
=x4/x2 + 2x2/x2
=x2+2
Therefore, By dividing x4 + 2x2 by x2 we get x2+2.
(vi) Given that, 5x3+ 25x2+30x by 5x
=5x3/5x+25x2/5x + 30x/5x
=x2 + 5x + 6
Therefore, By dividing 5x3+ 25x2+30x by 5x we get x2 + 5x + 6.