Question

a rectangle has a height of 4w³ and a width of 5w² − 3w − 4. express the area of the entire rectangle. your answer should be a polynomial in standard form. image of the rectangle divided into three parts with 4w³ as height and 5w², -3w, -4 as width segments area = box

Explanation:

Step1: Recall area formula for rectangle

The area \( A \) of a rectangle is given by the product of its height (length) and width, i.e., \( A=\text{height} \times \text{width} \). Here, height \( = 4w^{3}\) and width \( = 5w^{2}-3w - 4\), so we need to compute \( 4w^{3}\times(5w^{2}-3w - 4) \).

Step2: Distribute the monomial

Using the distributive property (also known as the distributive law of multiplication over addition/subtraction), we multiply \( 4w^{3}\) with each term inside the parentheses:

• Multiply \( 4w^{3}\) and \( 5w^{2}\): \( 4w^{3}\times5w^{2}=4\times5\times w^{3 + 2}=20w^{5}\) (using the rule of exponents \( a^{m}\times a^{n}=a^{m + n}\))
• Multiply \( 4w^{3}\) and \( - 3w\): \( 4w^{3}\times(-3w)=4\times(-3)\times w^{3+1}=-12w^{4}\)
• Multiply \( 4w^{3}\) and \( - 4\): \( 4w^{3}\times(-4)=-16w^{3}\)

Step3: Combine the terms

Combine the results from the previous step to form the polynomial: \( 20w^{5}-12w^{4}-16w^{3}\) (since there are no like terms to combine further, and we arrange the terms in descending order of exponents to get the standard form).

Answer:

\( 20w^{5}-12w^{4}-16w^{3} \)