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## Dividing Polynomials by Monomials Worksheet PDF

**I. Simplify the following polynomial divided by monomial:
**(i) 15m

^{3}+ 9m

^{2}+ 6m by 3m

(ii) 5x

^{3}+ 30x

^{2}+ 15x by 5x

(iii) 48x

^{3}– 16x

^{2}+ 80x by 16x

(iv) -3y

^{6}+ 6y

^{4}+ y

^{2}+ 4 by 2y

^{2 }(v) 14a

^{2}b – 16ab – 20ab

^{2}by 2ab

(vi) 14x

^{3}y

^{3}+ 21x

^{4}y

^{2}– 49x

^{2}y

^{4}by -7x

^{2}y

^{2}

**Solution:**

(i) Given that,

15m^{3} + 9m^{2} + 6m by 3m

=15m^{3} + 9m^{2} + 6m/3m

=15m^{3}/3m + 9m^{2}/3m + 6m/3m

=5m^{2} + 3m + 2

Therefore, By dividing 15m^{3} + 9m^{2} + 6m by 3m we get 5m^{2} + 3m + 2.

(ii) Given that,

5x^{3} + 30x^{2} + 15x by 5x

=5x^{3} + 30x^{2} + 15x/5x

=5x^{3}/5x + 30x^{2}/5x + 15x/5x

=x^{2} + 6x +3

Therefore, By dividing 5x^{3} + 30x^{2} + 15x by 5x we get x^{2} + 6x +3.

(iii) Given that, 48x^{3} – 16x^{2} + 80x by 16x

=48x^{3} – 16x^{2} + 80x/16x

=48x^{3}/16x -16x^{2}/16x + 80x/16x

=3x^{2}-x +5

Therefore, By dividing 48x^{3} – 16x^{2} + 80x by 16x we get 3x^{2}-x +5.

(iv) Given that, -3y^{6} + 6y^{4} + y^{2} + 4 by 2y^{2}

=-3y^{6} + 6y^{4} + y^{2} + 4/2y^{2}

=-3y^{6}/2y^{2} + 6y^{4}/2y^{2} + y^{2}/2y^{2} +4/2y^{2}

=-3/2y^{4} + 3y^{2} + 1/2 +2/y^{2}

Therefore, By dividing -3y^{6} + 6y^{4} + y^{2} + 4 by 2y^{2} we get -3/2y^{4} + 3y^{2} + 1/2 +2/y^{2}.

(v) Given that, 14a^{2}b – 16ab – 20ab^{2} by 2ab

=14a^{2}b – 16ab – 20ab^{2}/2ab

=14a^{2}b/2ab – 16ab/2ab – 20ab^{2}/2ab

=7a-8-10b

Therefore, By dividing 14a^{2}b – 16ab – 20ab^{2} by 2ab we get 7a-8-10b.

(vi) Given that, 14x^{3}y^{3} + 21x^{4}y^{2} – 49x^{2}y^{4} by -7x^{2}y^{2}

=14x^{3}y^{3}/-7x^{2}y^{2} + 21x^{4}y^{2}/-7x^{2}y^{2}-49x^{2}y^{4}/-7x^{2}y^{2}

=-2xy-3x^{2}+7y^{2}

Therefore, By dividing 14x^{3}y^{3} + 21x^{4}y^{2} – 49x^{2}y^{4} by -7x^{2}y^{2} we get -2xy-3x^{2}+7y^{2}.

**II. Solve the following by dividing the polynomial by a monomial:
**(i) (x

^{2}– 5xy) ÷ 2x

(ii) (3z

^{3}– 6z

^{2}+ 12z) ÷ 3z

(iii) (4m

^{6}– 3m

^{5}+ 8m

^{4}) ÷ m

^{2}

(iv) (8a

^{7}– 6a

^{6}+ 2a

^{4}) ÷ a

^{3}

(v) (12y

^{5}– 21y

^{4}) ÷ (-3y

^{3})

(vi) (36a

^{6}– 72a

^{5}) ÷ 9a

^{5 }(vii) (x

^{4}-3x

^{3}+4x

^{2}+2x) ÷x

^{2}

**Solution:**

(i) Given that, (x^{2} – 5xy) ÷ 2x

=x^{2}/2x-5xy/2x

=1/2x-5/2y

Therefore, By dividing (x^{2} – 5xy) ÷ 2x we get 1/2x-5/2y.

(ii) Given that, (3z^{3} – 6z^{2} + 12z) ÷ 3z

=3z^{3}/3z-6z^{2}/3z +12z/3z

=z^{2}-6z+4

Therefore, By dividing 3z^{3} – 6z^{2} + 12z by 3z we get z^{2}-6z+4.

(iii) Given that, (4m^{6} – 3m^{5} + 8m^{4}) ÷ m^{2}

=4m^{6}/m^{2}-3m^{5}/m^{2} + 8m^{4}/m^{2}

=4m^{4}-3m^{3} + 8m^{2}

Therefore, By dividing 4m^{6} – 3m^{5} + 8m^{4} with m^{2} we get 4m^{4}-3m^{3} + 8m^{2}.

(iv) Given that, (8a^{7} – 6a^{6} + 2a^{4}) ÷ a^{3}

=8a^{7}/a^{3} -6a^{6}/a3 + 2a^{4}/a^{3}

=8a^{4}-6a^{3}+2a

Therefore, By dividing 8a^{7} – 6a^{6} + 2a^{4} with a^{3} we get 8a^{4}-6a^{3}+2a.

(v) Given that, (12y^{5} – 21y^{4}) ÷ (-3y^{3})

=12y^{5} /-3y^{3} + 21y^{4}/3y^{3}

=-4y^{2} +7y

Therefore, By dividing 12y^{5} – 21y^{4} with -3y^{3} we get -4y^{2} +7y.

(vi) Given that, (36a^{6} – 72a^{5}) ÷ 9a^{5}

=36a^{6} /9a^{5} – 72a^{5}/9a^{5}

=4a-8

Therefore, By dividing 36a^{6} – 72a^{5} with 9a^{5} we get 4a-8.

(vii) Given that, (x^{4}-3x^{3}+4x^{2}+2x) ÷x

=x^{4}/x-3x^{3}/x+4x^{2}/x+2x/x

=x^{3}-3x^{2}+4x+2

Therefore, By dividing x^{4}-3x^{3}+4x^{2}+2x by x we get x^{3}-3x^{2}+4x+2.

**III.** **Divide the following polynomial by monomial and write the answer in simplest form:
**(i) 8a

^{3}– 48a

^{2}+ 64a by 8a

(ii) 18m

^{2}n

^{2}– 2mn

^{2}+ 6mn

^{3}by 2mn

(iii) 8a

^{2}b – 4ab

^{2}– 20ab by 4ab

(iv) 6x

^{4}– 3x

^{3}+ (3/2)x

^{2}by 3x

(v) x

^{4}+ 2x

^{2}by x

^{2 }(vi) 5x

^{3}+ 25x

^{2}+30x by 5x

**Solution:**

(i) Given that, 8a^{3} – 48a^{2} + 64a by 8a

=8a^{3}/8a-48a^{2}/8a + 64a/8a

=a^{2}-6a+8

Therefore, By dividing 8a^{3} – 48a^{2} + 64a by 8a we get a^{2}-6a+8.

(ii) Given that, 18m^{2}n^{2} – 2mn^{2} + 6mn^{3} by 2mn

=18m^{2}n^{2}/2mn-2mn^{2}/2mn + 6mn^{3}/2mn

=9mn-n+3n^{2
}Therefore, By dividing 18m^{2}n^{2} – 2mn^{2} + 6mn^{3} by 2mn we get 9mn-n+3n^{2}.

(iii) Given that, 8a^{2}b – 4ab^{2} – 20ab by 4ab

=8a^{2}b/4ab-4ab^{2}/4ab – 20ab/4ab

=2a-b-5

Therefore, By dividing 8a^{2}b – 4ab^{2} – 20ab by 4ab we get 2a-b-5.

(iv) Given that, 6x^{4} – 3x^{3} + (3/2)x^{2} by 3x

=6x^{4}/3x-3x^{3}/3x + (3/2)x^{2}/3x

=2x^{3} – x^{2} + 1/3x

Therefore, By dividing 6x^{4} – 3x^{3} + (3/2)x^{2} by 3x we get 2x^{3} – x^{2} + 1/3x.

(v) Given that, x^{4} + 2x^{2} by x^{2}

=x^{4}/x^{2} + 2x^{2}/x^{2}

=x^{2}+2

Therefore, By dividing x^{4} + 2x^{2} by x^{2 } we get x^{2}+2.

(vi) Given that, 5x^{3}+ 25x^{2}+30x by 5x

=5x^{3}/5x+25x^{2}/5x + 30x/5x

=x^{2} + 5x + 6

Therefore, By dividing 5x^{3}+ 25x^{2}+30x by 5x we get x^{2} + 5x + 6.

** **