Question

5.1 check
simplify the following.

1. $(6a^2 + 5a + 10) - (4a^2 + 6a + 12)$
2. $(7b^2 + 6b - 7) + (4b^2 - 2)$
3. $(x - 5)(x^2 + 2x - 3)$
4. $(x + 2)(x + 2)(x + 2)$
5. $(x - y)(x + y)(2x + y)$

Explanation:

Step1: Distribute the negative sign

$(6a^2 + 5a + 10) - (4a^2 + 6a + 12) = 6a^2 + 5a + 10 - 4a^2 - 6a - 12$

Step2: Combine like terms

$(6a^2 - 4a^2) + (5a - 6a) + (10 - 12) = 2a^2 - a - 2$

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Step1: Remove parentheses and group like terms

$(7b^2 + 6b - 7) + (4b^2 - 2) = 7b^2 + 4b^2 + 6b - 7 - 2$

Step2: Combine like terms

$(7b^2 + 4b^2) + 6b + (-7 - 2) = 11b^2 + 6b - 9$

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Step1: Distribute $(x-5)$ to each term

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

Step2: Expand each product

$= x^3 + 2x^2 - 3x - 5x^2 - 10x + 15$

Step3: Combine like terms

$x^3 + (2x^2 - 5x^2) + (-3x - 10x) + 15 = x^3 - 3x^2 - 13x + 15$

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Step1: Rewrite as squared binomial first

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

Step2: Expand $(x+2)^2$

$(x^2 + 4x + 4)(x+2)$

Step3: Distribute and expand

$x(x^2 + 4x + 4) + 2(x^2 + 4x + 4) = x^3 + 4x^2 + 4x + 2x^2 + 8x + 8$

Step4: Combine like terms

$x^3 + (4x^2 + 2x^2) + (4x + 8x) + 8 = x^3 + 6x^2 + 12x + 8$

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Step1: Use difference of squares first

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

Step2: Distribute $(x^2 - y^2)$

$x^2(2x+y) - y^2(2x+y) = 2x^3 + x^2y - 2xy^2 - y^3$

Step3: Arrange terms (optional, standard form)

$2x^3 + x^2y - 2xy^2 - y^3$

Answer:

1. $2a^2 - a - 2$
2. $11b^2 + 6b - 9$
3. $x^3 - 3x^2 - 13x + 15$
4. $x^3 + 6x^2 + 12x + 8$
5. $2x^3 + x^2y - 2xy^2 - y^3$