The length of a rectangle is represented by the polynomial 2x^3 - 5x^2 + 8 and the width is represented by the polynomial x + 3.

Question

Complete the following statements about the polynomial that represents the area of the rectangle.
The polynomial has ___ terms.
The constant term of the polynomial is ___.
The polynomial has a degree of ___.
The leading coefficient of the polynomial is ___.

Answer 1

To find the area of the rectangle represented by the given polynomials for length and width, we need to multiply them:

Length: \( L = 2x^3 - 5x^2 + 8 \)
Width: \( W = x + 3 \)

The area \( A \) is given by:

\[ A = L \times W = (2x^3 - 5x^2 + 8)(x + 3) \]

We can expand this using the distributive property (also known as the FOIL method in this case):

\[ A = (2x^3)(x) + (2x^3)(3) + (-5x^2)(x) + (-5x^2)(3) + (8)(x) + (8)(3) \]

Calculating each term:

\[ = 2x^4 + 6x^3 - 5x^3 - 15x^2 + 8x + 24 \]

Now, combine like terms:

\[ = 2x^4 + (6x^3 - 5x^3) - 15x^2 + 8x + 24 \] \[ = 2x^4 + 1x^3 - 15x^2 + 8x + 24 \]

Now we can answer the statements.

1. The polynomial has 5 terms (2x^4, x^3, -15x^2, 8x, and 24).
2. The constant term of the polynomial is 24.
3. The polynomial has a degree of 4 (the highest power of x).
4. The leading coefficient of the polynomial is 2 (the coefficient of the term with the highest degree, which is 2x^4).

So the completed statements are:

• The polynomial has 5 terms.
• The constant term of the polynomial is 24.
• The polynomial has a degree of 4.
• The leading coefficient of the polynomial is 2.