Question

simplify each problem by performing the indicated operations. write answers in descending order.
1.) $(-5x)(x^{4})(4x^{2})$
2.) $3x^{5}(2x^{2}-5)$
3.) $(x + 2)(3x - 2)$
4.) $(x + 5)(x^{2}-4x + 5)$
5.) $(x + 2)(x - 1)(x + 3)$
6.) $(3x^{2}+5x - 8)-(x^{2}+5x - 2)$
7.) $5(3a^{2}+2)-2(a^{2}-4)$
8.) $-(7x^{2}+x - 8)-2(5x^{2}+6)$
9.) $(3x - 2)^{2}$
10.) $(x - 2)^{3}$

Explanation:

1.) Step1: Multiply coefficients first

$(-5) \times 1 \times 4 = -20$

1.) Step2: Multiply variable terms

$x \times x^4 \times x^2 = x^{1+4+2} = x^7$

1.) Step3: Combine results

$-20 \times x^7$

2.) Step1: Distribute $3x^5$ to each term

$3x^5 \times 2x^2 - 3x^5 \times 5$

2.) Step2: Simplify each product

$6x^{5+2} - 15x^5 = 6x^7 - 15x^5$

3.) Step1: Apply distributive property (FOIL)

$x(3x) + x(-2) + 2(3x) + 2(-2)$

3.) Step2: Simplify each term

$3x^2 - 2x + 6x - 4$

3.) Step3: Combine like terms

$3x^2 + 4x - 4$

4.) Step1: Distribute $x$ and $5$

$x(x^2 - 4x + 5) + 5(x^2 - 4x + 5)$

4.) Step2: Expand each product

$x^3 - 4x^2 + 5x + 5x^2 - 20x + 25$

4.) Step3: Combine like terms

$x^3 + x^2 - 15x + 25$

5.) Step1: Multiply first two binomials

$(x+2)(x-1) = x^2 - x + 2x - 2 = x^2 + x - 2$

5.) Step2: Multiply by third binomial

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

5.) Step3: Distribute each term

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

5.) Step4: Simplify and combine like terms

$x^3 + 3x^2 + x^2 + 3x - 2x - 6 = x^3 + 4x^2 + x - 6$

6.) Step1: Remove parentheses (distribute negative)

$3x^2 + 5x - 8 - x^2 - 5x + 2$

6.) Step2: Combine like terms

$(3x^2 - x^2) + (5x - 5x) + (-8 + 2) = 2x^2 - 6$

7.) Step1: Distribute coefficients to each term

$5(3a^2) + 5(2) - 2(a^2) + 2(4)$

7.) Step2: Simplify each product

$15a^2 + 10 - 2a^2 + 8$

7.) Step3: Combine like terms

$(15a^2 - 2a^2) + (10 + 8) = 13a^2 + 18$

8.) Step1: Distribute negative signs

$-7x^2 - x + 8 - 10x^2 - 12$

8.) Step2: Combine like terms

$(-7x^2 - 10x^2) + (-x) + (8 - 12) = -17x^2 - x - 4$

9.) Step1: Expand using square formula

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

9.) Step2: Simplify each term

$9x^2 - 12x + 4$

10.) Step1: Expand as $(x-2)(x-2)^2$

First expand $(x-2)^2 = x^2 - 4x + 4$

10.) Step2: Multiply by $(x-2)$

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

10.) Step3: Expand and combine like terms

$x^3 - 4x^2 + 4x - 2x^2 + 8x - 8 = x^3 - 6x^2 + 12x - 8$

Answer:

1.) $\boldsymbol{-20x^7}$
2.) $\boldsymbol{6x^7 - 15x^5}$
3.) $\boldsymbol{3x^2 + 4x - 4}$
4.) $\boldsymbol{x^3 + x^2 - 15x + 25}$
5.) $\boldsymbol{x^3 + 4x^2 + x - 6}$
6.) $\boldsymbol{2x^2 - 6}$
7.) $\boldsymbol{13a^2 + 18}$
8.) $\boldsymbol{-17x^2 - x - 4}$
9.) $\boldsymbol{9x^2 - 12x + 4}$
10.) $\boldsymbol{x^3 - 6x^2 + 12x - 8}$