Use the FOIL method to find the product below. $(x+5)(x^2-3x)$
A. $x^3+5x^2-15x$
B. $x^3+2x^2-15$
C. $x^3+5x^2-15$
D. $x^3+2x^2-15x$
Answer (1)
Multiply the First terms: x × x 2 = x 3 .
Multiply the Outer terms: x × ( − 3 x ) = − 3 x 2 .
Multiply the Inner terms: 5 × x 2 = 5 x 2 .
Multiply the Last terms: 5 × ( − 3 x ) = − 15 x .
Combine like terms: x 3 − 3 x 2 + 5 x 2 − 15 x = x 3 + 2 x 2 − 15 x .
The final answer is x 3 + 2 x 2 − 15 x .
Explanation
Understanding the Problem We are asked to multiply ( x + 5 ) ( x 2 − 3 x ) using the FOIL method. FOIL stands for First, Outer, Inner, Last, which describes the order in which we multiply the terms of the two expressions.
Multiplying the First Terms First, we multiply the first terms of each expression: x ⋅ x 2 = x 3 .
Multiplying the Outer Terms Next, we multiply the outer terms: x ⋅ ( − 3 x ) = − 3 x 2 .
Multiplying the Inner Terms Then, we multiply the inner terms: 5 ⋅ x 2 = 5 x 2 .
Multiplying the Last Terms Finally, we multiply the last terms: 5 ⋅ ( − 3 x ) = − 15 x .
Combining the Terms Now, we add all the terms together: x 3 − 3 x 2 + 5 x 2 − 15 x .
Simplifying the Expression Combine like terms: x 3 + ( − 3 + 5 ) x 2 − 15 x = x 3 + 2 x 2 − 15 x .
Final Answer Therefore, the product of ( x + 5 ) ( x 2 − 3 x ) is x 3 + 2 x 2 − 15 x . The correct answer is D.
Examples
Understanding how to expand polynomial expressions like this is fundamental in many areas, such as physics, engineering, and computer graphics. For example, when calculating the trajectory of a projectile, you might need to expand expressions involving time and initial velocity to model the object's position. Similarly, in computer graphics, expanding polynomial expressions can help determine the shape and appearance of curves and surfaces.
