Question

given the binomial z = 2x - 3, find z³ - 5z + 7.
a. 8x³ - 10x - 5
b. 8x³ - 24x² + 8x - 35
c. 8x³ - 36x² + 44x - 5
d. 8x³ - 12x² - 28x - 35

Explanation:

Step1: Expand \( z^3 \) where \( z = 2x - 3 \)

Using the formula \( (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \), here \( a = 2x \), \( b = 3 \).
\( z^3=(2x - 3)^3=(2x)^3-3(2x)^2(3)+3(2x)(3)^2-3^3 \)
\( = 8x^3 - 36x^2 + 54x - 27 \)

Step2: Expand \( -5z \)

Substitute \( z = 2x - 3 \), so \( -5z=-5(2x - 3)=-10x + 15 \)

Step3: Combine \( z^3 - 5z + 7 \)

Add the expanded \( z^3 \), \( -5z \), and \( 7 \):
\( (8x^3 - 36x^2 + 54x - 27)+(-10x + 15)+7 \)
Combine like terms:

• \( x^3 \): \( 8x^3 \)
• \( x^2 \): \( -36x^2 \)
• \( x \): \( 54x - 10x = 44x \)
• Constants: \( -27 + 15 + 7 = -5 \)

Result: \( 8x^3 - 36x^2 + 44x - 5 \)

Answer:

C. \( 8x^3 - 36x^2 + 44x - 5 \)