Question

multiply and simplify each expression below: a) $(3x + 8)(7x^{2}-5x - 2)$ b) $(3x^{5}-9y^{4})^{2}$

Explanation:

Step1: Expand the first - expression using distributive property

\[

$$\begin{align*} &(3x + 8)(7x^{2}-5x - 2)\\ =&3x(7x^{2}-5x - 2)+8(7x^{2}-5x - 2)\\ =&(3x\times7x^{2}-3x\times5x - 3x\times2)+(8\times7x^{2}-8\times5x - 8\times2)\\ =&21x^{3}-15x^{2}-6x + 56x^{2}-40x - 16 \end{align*}$$

\]

Step2: Combine like - terms for the first expression

\[

$$\begin{align*} &21x^{3}+(-15x^{2}+56x^{2})+(-6x - 40x)-16\\ =&21x^{3}+41x^{2}-46x - 16 \end{align*}$$

\]

Step3: Expand the second expression using the formula \((a - b)^{2}=a^{2}-2ab + b^{2}\)

\[

$$\begin{align*} &(3x^{5}-9y^{4})^{2}\\ =&(3x^{5})^{2}-2\times(3x^{5})\times(9y^{4})+(9y^{4})^{2}\\ =&9x^{10}-54x^{5}y^{4}+81y^{8} \end{align*}$$

\]

Answer:

a) \(21x^{3}+41x^{2}-46x - 16\)
b) \(9x^{10}-54x^{5}y^{4}+81y^{8}\)