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Simplify the following algebraic expressions by removing brackets and collecting like terms: a) 4a2(2a+b) - 3a2(5a+2b), b) 3m(m2 - 2n2) + 6m(m2 - 2n2), c) 4y2(x2 - 4xy + y2) - 4y2(x2 + 4xy - y2).

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6 parts
a

−7a3−2a2b-7a^3 - 2a^2b

b

9m3−18mn29m^3 - 18mn^2

c

−32xy3+8y4-32xy^3 + 8y^4

d

14x3−10x2+22x14x^3 - 10x^2 + 22x

e

10p2b2−12b410p^2b^2 - 12b^4

f

9k + 5

Step-by-step solution

12 steps

Here are the solutions for removing the brackets and collecting like terms.

Part a· 2 steps

Remove the brackets and collect the like terms for 4a2(2a+b)−3a2(5a+2b)4a^2(2a+b) - 3a^2(5a+2b).

  1. Distribute 4a24a^2 and −3a2-3a^2 into their respective parentheses.

    4a2(2a+b)−3a2(5a+2b)=(4a2⋅2a)+(4a2⋅b)−(3a2⋅5a)−(3a2⋅2b)4a^2(2a+b) - 3a^2(5a+2b) = (4a^2 \cdot 2a) + (4a^2 \cdot b) - (3a^2 \cdot 5a) - (3a^2 \cdot 2b) =8a3+4a2b−15a3−6a2b= 8a^3 + 4a^2b - 15a^3 - 6a^2b

  2. Collect like terms.

    =(8a3−15a3)+(4a2b−6a2b)= (8a^3 - 15a^3) + (4a^2b - 6a^2b) =−7a3−2a2b= -7a^3 - 2a^2b

    The simplified expression is −7a3−2a2b\boxed{-7a^3 - 2a^2b}.

Part b

9m3−18mn29m^3 - 18mn^2

Remove the brackets and collect the like terms for 3m(m2−2n2)+6m(m2−2n2)3m(m^2 - 2n^2) + 6m(m^2 - 2n^2).

  1. Distribute $3m$ and $6m$ into their respective parentheses.

    3m(m2−2n2)+6m(m2−2n2)=(3m⋅m2)−(3m⋅2n2)+(6m⋅m2)−(6m⋅2n2)3m(m^2 - 2n^2) + 6m(m^2 - 2n^2) = (3m \cdot m^2) - (3m \cdot 2n^2) + (6m \cdot m^2) - (6m \cdot 2n^2) =3m3−6mn2+6m3−12mn2= 3m^3 - 6mn^2 + 6m^3 - 12mn^2

  2. Collect like terms.

    =(3m3+6m3)+(−6mn2−12mn2)= (3m^3 + 6m^3) + (-6mn^2 - 12mn^2) =9m3−18mn2= 9m^3 - 18mn^2

    The simplified expression is 9m3−18mn2\boxed{9m^3 - 18mn^2}.

Part c

−32xy3+8y4-32xy^3 + 8y^4

Remove the brackets and collect the like terms for 4y2(x2−4xy+y2)−4y2(x2+4xy−y2)4y^2(x^2 - 4xy + y^2) - 4y^2(x^2 + 4xy - y^2).

  1. Distribute 4y24y^2 and −4y2-4y^2 into their respective parentheses.

    4y2(x2−4xy+y2)−4y2(x2+4xy−y2)4y^2(x^2 - 4xy + y^2) - 4y^2(x^2 + 4xy - y^2) =(4y2⋅x2)−(4y2⋅4xy)+(4y2⋅y2)−(4y2⋅x2)−(4y2⋅4xy)+(4y2⋅y2)= (4y^2 \cdot x^2) - (4y^2 \cdot 4xy) + (4y^2 \cdot y^2) - (4y^2 \cdot x^2) - (4y^2 \cdot 4xy) + (4y^2 \cdot y^2) =4x2y2−16xy3+4y4−4x2y2−16xy3+4y4= 4x^2y^2 - 16xy^3 + 4y^4 - 4x^2y^2 - 16xy^3 + 4y^4

  2. Collect like terms.

    =(4x2y2−4x2y2)+(−16xy3−16xy3)+(4y4+4y4)= (4x^2y^2 - 4x^2y^2) + (-16xy^3 - 16xy^3) + (4y^4 + 4y^4) =0−32xy3+8y4= 0 - 32xy^3 + 8y^4 =−32xy3+8y4= -32xy^3 + 8y^4

    The simplified expression is −32xy3+8y4\boxed{-32xy^3 + 8y^4}.

Part d

14x3−10x2+22x14x^3 - 10x^2 + 22x

Remove the brackets and collect the like terms for 4x(x2−4x+6)+2x(5x2+3x−1)4x(x^2 - 4x + 6) + 2x(5x^2 + 3x - 1).

  1. Distribute $4x$ and $2x$ into their respective parentheses.

    4x(x2−4x+6)+2x(5x2+3x−1)4x(x^2 - 4x + 6) + 2x(5x^2 + 3x - 1) =(4x⋅x2)−(4x⋅4x)+(4x⋅6)+(2x⋅5x2)+(2x⋅3x)−(2x⋅1)= (4x \cdot x^2) - (4x \cdot 4x) + (4x \cdot 6) + (2x \cdot 5x^2) + (2x \cdot 3x) - (2x \cdot 1) =4x3−16x2+24x+10x3+6x2−2x= 4x^3 - 16x^2 + 24x + 10x^3 + 6x^2 - 2x

  2. Collect like terms.

    =(4x3+10x3)+(−16x2+6x2)+(24x−2x)= (4x^3 + 10x^3) + (-16x^2 + 6x^2) + (24x - 2x) =14x3−10x2+22x= 14x^3 - 10x^2 + 22x

    The simplified expression is 14x3−10x2+22x\boxed{14x^3 - 10x^2 + 22x}.

Part e

10p2b2−12b410p^2b^2 - 12b^4

Remove the brackets and collect the like terms for p2(2p2−5b2)−2(p4+3b4)+3b2(5p2−2b2)p^2(2p^2 - 5b^2) - 2(p^4 + 3b^4) + 3b^2(5p^2 - 2b^2).

  1. Distribute p2p^2, −2-2, and 3b23b^2 into their respective parentheses.

    p2(2p2−5b2)−2(p4+3b4)+3b2(5p2−2b2)p^2(2p^2 - 5b^2) - 2(p^4 + 3b^4) + 3b^2(5p^2 - 2b^2) =(p2⋅2p2)−(p2⋅5b2)−(2⋅p4)−(2⋅3b4)+(3b2⋅5p2)−(3b2⋅2b2)= (p^2 \cdot 2p^2) - (p^2 \cdot 5b^2) - (2 \cdot p^4) - (2 \cdot 3b^4) + (3b^2 \cdot 5p^2) - (3b^2 \cdot 2b^2) =2p4−5p2b2−2p4−6b4+15p2b2−6b4= 2p^4 - 5p^2b^2 - 2p^4 - 6b^4 + 15p^2b^2 - 6b^4

  2. Collect like terms.

    =(2p4−2p4)+(−5p2b2+15p2b2)+(−6b4−6b4)= (2p^4 - 2p^4) + (-5p^2b^2 + 15p^2b^2) + (-6b^4 - 6b^4) =0+10p2b2−12b4= 0 + 10p^2b^2 - 12b^4 =10p2b2−12b4= 10p^2b^2 - 12b^4

    The simplified expression is 10p2b2−12b4\boxed{10p^2b^2 - 12b^4}.

Part f9k + 5

Remove the brackets and collect the like terms for 10(k−15)+8(k+12)−9(k−13)10(k - \frac{1}{5}) + 8(k + \frac{1}{2}) - 9(k - \frac{1}{3}).

  1. Distribute 10, 8, and −9-9 into their respective parentheses.

    10(k−15)+8(k+12)−9(k−13)10(k - \frac{1}{5}) + 8(k + \frac{1}{2}) - 9(k - \frac{1}{3}) =(10⋅k)−(10⋅15)+(8⋅k)+(8⋅12)−(9⋅k)−(9⋅−13)= (10 \cdot k) - (10 \cdot \frac{1}{5}) + (8 \cdot k) + (8 \cdot \frac{1}{2}) - (9 \cdot k) - (9 \cdot -\frac{1}{3}) =10k−105+8k+82−9k+93= 10k - \frac{10}{5} + 8k + \frac{8}{2} - 9k + \frac{9}{3} =10k−2+8k+4−9k+3= 10k - 2 + 8k + 4 - 9k + 3

  2. Collect like terms.

    =(10k+8k−9k)+(−2+4+3)= (10k + 8k - 9k) + (-2 + 4 + 3) =(18k−9k)+(2+3)= (18k - 9k) + (2 + 3) =9k+5= 9k + 5

    The simplified expression is 9k+5\boxed{9k + 5}.

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