Question

expand the polynomial $(5x - 1)^3$ by multiplication.

select one:
a. $125x^3 - 1$
b. $125x^3 - 75x^2 + 15x - 1$
c. $125x^3 + 75x^2 + 15x + 1$
d. $125x^3 - 25x^2 - 5x + 1$

Explanation:

Step1: Rewrite as product of binomials

$(5x-1)^3=(5x-1)(5x-1)(5x-1)$

Step2: Multiply first two binomials

First, calculate $(5x-1)(5x-1)$:

$$\begin{align*} (5x-1)(5x-1)&=5x\cdot5x + 5x\cdot(-1) + (-1)\cdot5x + (-1)\cdot(-1)\\ &=25x^2 -5x -5x +1\\ &=25x^2 -10x +1 \end{align*}$$

Step3: Multiply result by third binomial

Multiply $(25x^2 -10x +1)$ by $(5x-1)$:

$$\begin{align*} &(25x^2 -10x +1)(5x-1)\\ &=25x^2\cdot5x +25x^2\cdot(-1) -10x\cdot5x -10x\cdot(-1) +1\cdot5x +1\cdot(-1)\\ &=125x^3 -25x^2 -50x^2 +10x +5x -1\\ &=125x^3 -75x^2 +15x -1 \end{align*}$$

Answer:

B. $125x^3 -75x^2 +15x -1$