Question

name teacher hour

1. multiply (x + 2) (x² - 3x - 2)

1. a rectangular swimming pool has a length of (3x² - 17x + 2) feet and a width of x feet.

write a polynomial for the area of the pool and find the area of the pool when the width is 15
feet.

polynomial for the area =

area =

1. fill in pascal’s triangle to row 6

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Explanation:

Question 13: Multiply \((x + 2)(x^2 - 3x - 2)\)

Step 1: Apply the distributive property (FOIL for polynomials)

Multiply \(x\) by each term in \((x^2 - 3x - 2)\) and \(2\) by each term in \((x^2 - 3x - 2)\):
\(x(x^2 - 3x - 2) = x^3 - 3x^2 - 2x\)
\(2(x^2 - 3x - 2) = 2x^2 - 6x - 4\)

Step 2: Combine like terms

Add the two results together:
\((x^3 - 3x^2 - 2x) + (2x^2 - 6x - 4) = x^3 + (-3x^2 + 2x^2) + (-2x - 6x) - 4\)
Simplify the like terms:
\(x^3 - x^2 - 8x - 4\)

Step 1: Recall the formula for the area of a rectangle

The area \(A\) of a rectangle is given by \(A = \text{length} \times \text{width}\).
Given length \(= (3x^2 - 17x + 2)\) feet and width \(= x\) feet.

Step 2: Multiply the length and width to get the area polynomial

\(A = x(3x^2 - 17x + 2) = 3x^3 - 17x^2 + 2x\)

Step 3: Find the area when \(x = 15\)

Substitute \(x = 15\) into the polynomial \(3x^3 - 17x^2 + 2x\):
First, calculate each term:
\(3(15)^3 = 3 \times 3375 = 10125\)
\(-17(15)^2 = -17 \times 225 = -3825\)
\(2(15) = 30\)
Then, add the terms together:
\(10125 - 3825 + 30 = 6330\)

Step 1: Recall the rules of Pascal’s triangle

• The first and last element of each row is \(1\).
• Each element (except the first and last) is the sum of the two elements directly above it.

Step 2: Build Pascal’s triangle row by row

• Row 0: \(1\)
• Row 1: \(1, 1\)
• Row 2: \(1, 2, 1\) ( \(1 + 1 = 2\))
• Row 3: \(1, 3, 3, 1\) ( \(1 + 2 = 3\), \(2 + 1 = 3\))
• Row 4: \(1, 4, 6, 4, 1\) ( \(1 + 3 = 4\), \(3 + 3 = 6\), \(3 + 1 = 4\))
• Row 5: \(1, 5, 10, 10, 5, 1\) ( \(1 + 4 = 5\), \(4 + 6 = 10\), \(6 + 4 = 10\), \(4 + 1 = 5\))
• Row 6: \(1, 6, 15, 20, 15, 6, 1\) ( \(1 + 5 = 6\), \(5 + 10 = 15\), \(10 + 10 = 20\), \(10 + 5 = 15\), \(5 + 1 = 6\))

Answer:

\(x^3 - x^2 - 8x - 4\)

Question 14: Area of a rectangular swimming pool