Question

The length, width, and height of a rectangular box are represented by (x + 7), (x - 3), and (x + 5) respectively. When the volume is expressed as a polynomial in standard form, what is the coefficient of the 2nd term?

Answer 1

To find the coefficient of the second term for the volume of the rectangular box, we multiplied the expressions for each dimension, combined like terms, and found that the coefficient is 9.

Step-by-step explanation:

To express the volume of a rectangular box as a polynomial when the sides are represented by algebraic expressions, the volume (V) is found by multiplying the length, width, and height together.

In this case, the expressions for the length, width, and height are (x + 7), (x - 3), and (x + 5) respectively. The volume in polynomial form would be:

• (x + 7) \(\times\) (x - 3) \(\times\) (x + 5)

First, multiply the terms in the first two parentheses:

• (x \(\times\) x + x \(\times\) -3 + 7 \(\times\) x - 21)

This simplifies to:

• x2 + 4x - 21

Now multiply this result by the third term, (x + 5), to find the expanded polynomial expression:

• (x2 + 4x - 21) \(\times\) (x + 5)

Which gives:

• x3 + 5x2 + 4x2 + 20x - 21x - 105

Combining like terms, the volume can be expressed as:

• x3 + 9x2 + [term]x - 105

The coefficient of the second term of this polynomial is 9.