How to expand this?

Question

How would I expand this equation?

(2x - 3)⁵

Answer 1

Use the Binomial Theorem

(2x - 3)⁵ = 1(2x)⁵ + 5(2x)⁴(-3) + 10(2x)³(-3)² + 10(2x)²(-3)³ + 5(2x)(-3)⁴ + (-3)⁵

Answer 2

Hi Kana! Our first step is going to be breaking up the expression so that we can see the steps more easily. (Remember, when you multiply to exponentiation terms, the powers add).

(2x - 3)⁵ = (2x - 3)² * (2x - 3)² * (2x -3)

Now we can see that our first step is to find expand (2x - 3)². Remember, when multiplying two polynomials, every term in the first must be multiplied by every term in the second and then you add/subtract like terms to simplify.

(2x - 3)² = (2x -3) * (2x - 3) = 4x² -6x - 6x +9 = 4x² - 12x +9

Now we plug that polynomial in for both the (2x - 3)² terms. This saves us a step.

(2x - 3)² * (2x - 3)² * (2x -3) = (4x² - 12x +9) * (4x² - 12x +9) * (2x -3)

Next, multiply the two (4x² - 12x +9) terms together.

(4x² - 12x +9) * (4x² - 12x +9) = 16x⁴ - 48x³ + 36x² -48x³ + 144x² - 108x + 36x² -108x +81

= 16x⁴ - 96x³ + 216x² - 216x +81

Plug this in to our full equation and expand one more time.

(16x⁴ - 96x³ +216x² - 216x +81)(2x-3)

= 32x⁵ - 48x⁴ - 192x⁴ + 288x³ + 432x³ - 628x² - 432x^{2 +} 648x + 162x - 243

= 32x⁵ - 240x⁴ +720x³ - 1080x² + 810x - 243

I hope this helps! Please reach out if you have any follow up questions.