Which of these is a correct expansion of $(3 x-1)(4 x^2+5)$?
A. $3 x \cdot 4 x^2+(-1) \cdot 4 x^2+4 x^2 \cdot 5+(-1) \cdot 5$
B. $3 x \cdot 4 x^2+3 x \cdot 5+(-1) \cdot 4 x^2+(-1) \cdot 5$
C. $3 x \cdot 4 x^2+3 x \cdot 5+1 \cdot 4 x^2+1 \cdot 5$
Answer (1)
Apply the distributive property to expand ( 3 x − 1 ) ( 4 x 2 + 5 ) .
The expansion is 3 x ⋅ 4 x 2 + 3 x ⋅ 5 + ( − 1 ) ⋅ 4 x 2 + ( − 1 ) ⋅ 5 .
Compare the expansion with the given options.
The correct expansion is B .
Explanation
Understanding the Problem We need to expand the expression ( 3 x − 1 ) ( 4 x 2 + 5 ) and identify the correct expansion from the given options.
Expanding the Expression To expand the expression, we use the distributive property (also known as the FOIL method). This means we multiply each term in the first parentheses by each term in the second parentheses:
( 3 x − 1 ) ( 4 x 2 + 5 ) = 3 x ⋅ 4 x 2 + 3 x ⋅ 5 + ( − 1 ) ⋅ 4 x 2 + ( − 1 ) ⋅ 5
Comparing with Options Now, let's compare the expansion we obtained with the given options:
Option A: 3 x ⋅ 4 x 2 + ( − 1 ) ⋅ 4 x 2 + 4 x 2 ⋅ 5 + ( − 1 ) ⋅ 5 (Incorrect) Option B: 3 x ⋅ 4 x 2 + 3 x ⋅ 5 + ( − 1 ) ⋅ 4 x 2 + ( − 1 ) ⋅ 5 (Correct) Option C: 3 x ⋅ 4 x 2 + 3 x ⋅ 5 + 1 ⋅ 4 x 2 + 1 ⋅ 5 (Incorrect)
Identifying the Correct Option The correct expansion is option B.
Examples
Understanding polynomial expansion is crucial in various fields, such as physics and engineering, where complex equations often need simplification. For instance, when calculating the trajectory of a projectile, you might encounter an expression like ( v 0 + a t ) ( t ) , where v 0 is the initial velocity, a is the acceleration, and t is the time. Expanding this expression gives v 0 t + a t 2 , which helps in analyzing the projectile's displacement over time. This skill is also fundamental in computer graphics for rendering and manipulating 3D models, where polynomial expressions are used to define curves and surfaces.
