Question

names:
3.5.1 practice problems
simplify the expressions below. write your answer in standard form.

1. $(6x^2 + 7x)+(5 - 2x^2 + 9x)$
2. $(8a^2 - 6 - 8a)+(1 - 6a - 7a^2)$
3. $(-5x - 2x^3 - 4x^2)+(x^3 - 7x + 4x^2)$
4. $(a^4 + 10a^2 - 9a)-(2a^2 - 3a^4 + a - 8a^3)$
5. $(8x^3 - 6x^4 + 3)-(3x^3 - 3 + 8x^4)$
6. $(2a^2b - 5b^3 + 4a^4b^2)-(7b^3 + 8a^4b^2 - 7a^2b)$
7. $(x + 2)(2x^2 - x - 9)$
8. $m^2(m - 1)^3$

Explanation:

Step1: Remove parentheses, combine like terms

$(6x^2 + 7x) + (5 - 2x^2 + 9x) = 6x^2 - 2x^2 + 7x + 9x + 5$

Step2: Calculate each like term group

$= (6-2)x^2 + (7+9)x + 5 = 4x^2 + 16x + 5$

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Step1: Remove parentheses, combine like terms

$(8a^2 - 6 - 8a) + (1 - 6a - 7a^2) = 8a^2 - 7a^2 -8a -6a -6 + 1$

Step2: Calculate each like term group

$= (8-7)a^2 + (-8-6)a + (-6+1) = a^2 -14a -5$

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Step1: Remove parentheses, combine like terms

$(-5x - 2x^3 - 4x^2) + (x^3 -7x + 4x^2) = -2x^3 + x^3 -4x^2 + 4x^2 -5x -7x$

Step2: Calculate each like term group

$= (-2+1)x^3 + (-4+4)x^2 + (-5-7)x = -x^3 -12x$

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Step1: Remove parentheses, combine like terms

$(a^4 +10a^2 -9a) - (2a^2 -3a^4 +a -8a^3) = a^4 +3a^4 +8a^3 +10a^2 -2a^2 -9a -a$

Step2: Calculate each like term group

$= (1+3)a^4 +8a^3 + (10-2)a^2 + (-9-1)a = 4a^4 +8a^3 +8a^2 -10a$

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Step1: Remove parentheses, combine like terms

$(8x^3 -6x^4 +3) - (3x^3 -3 +8x^4) = -6x^4 -8x^4 +8x^3 -3x^3 +3 +3$

Step2: Calculate each like term group

$= (-6-8)x^4 + (8-3)x^3 + (3+3) = -14x^4 +5x^3 +6$

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Step1: Remove parentheses, combine like terms

$(2a^2b -5b^3 +4a^4b^2) - (7b^3 +8a^4b^2 -7a^2b) = 4a^4b^2 -8a^4b^2 +2a^2b +7a^2b -5b^3 -7b^3$

Step2: Calculate each like term group

$= (4-8)a^4b^2 + (2+7)a^2b + (-5-7)b^3 = -4a^4b^2 +9a^2b -12b^3$

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Step1: Expand using distributive property

$(x+2)(2x^2 -x -9) = x(2x^2 -x -9) + 2(2x^2 -x -9)$

Step2: Distribute terms

$= 2x^3 -x^2 -9x +4x^2 -2x -18$

Step3: Combine like terms

$= 2x^3 + (-1+4)x^2 + (-9-2)x -18 = 2x^3 +3x^2 -11x -18$

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Step1: Expand $(m-1)^3$ first

$(m-1)^3 = m^3 -3m^2 +3m -1$

Step2: Multiply by $m^2$

$m^2(m^3 -3m^2 +3m -1) = m^2 \cdot m^3 -m^2 \cdot 3m^2 +m^2 \cdot 3m -m^2 \cdot 1$

Step3: Simplify each term

$= m^5 -3m^4 +3m^3 -m^2$

Answer:

1. $4x^2 + 16x + 5$
2. $a^2 - 14a - 5$
3. $-x^3 - 12x$
4. $4a^4 + 8a^3 + 8a^2 - 10a$
5. $-14x^4 + 5x^3 + 6$
6. $-4a^4b^2 + 9a^2b - 12b^3$
7. $2x^3 + 3x^2 - 11x - 18$
8. $m^5 - 3m^4 + 3m^3 - m^2$