Question

mixed review
assessment readiness

1. find the standard form for the product of (x² + 8) and (x² − 2). choose true or false for each statement about the product.

a. it is a 4th degree polynomial. ⓐ true ○ false
b. the constant term is −16. ⓑ true ○ false
c. it has 3 terms. ⓒ true ○ false

1. multiply (5x− 9)². choose true or false for each statement about the product.

a. the coefficient of the x-term is −45. ○ true ⓒ false
b. the leading term is 25x². ⓑ true ○ false
c. the constant term is 81. ⓒ true ○ false

1. find the product (3x + 6)(3x − 6). show your work.

(3x+6)(3x-6)
9x² -x -36 (crossed out)
9x² +18x +18 (partial)
9x² -36 (correct)

1. find the product (x + 10)(4x + 5). show your work.

4x² +5x +40x +50 (work)

1. a rectangle has a length (x + 6) m and a width of 7 m. write expressions for the perimeter and area of the rectangle. explain how you determined your...

Explanation:

Question 3:

Step1: Identify the formula

We can use the difference of squares formula: \((a + b)(a - b)=a^{2}-b^{2}\). Here, \(a = 3x\) and \(b = 6\).

Step2: Apply the formula

Substitute \(a = 3x\) and \(b = 6\) into the formula: \((3x)^{2}-6^{2}\).

Step3: Simplify the terms

Calculate \((3x)^{2}=9x^{2}\) and \(6^{2} = 36\). So the product is \(9x^{2}-36\).

Step1: Use the distributive property (FOIL method)

Multiply each term in the first binomial by each term in the second binomial: \((x+10)(4x + 5)=x\times(4x)+x\times5+10\times(4x)+10\times5\).

Step2: Simplify each product

\(x\times(4x)=4x^{2}\), \(x\times5 = 5x\), \(10\times(4x)=40x\), \(10\times5 = 50\).

Step3: Combine like terms

Combine the \(x\)-terms: \(5x+40x = 45x\). So the product is \(4x^{2}+45x + 50\).

Step1: Recall the formulas for perimeter and area of a rectangle

The perimeter \(P\) of a rectangle is given by \(P = 2\times(\text{length}+\text{width})\) and the area \(A\) is given by \(A=\text{length}\times\text{width}\).

Step2: Calculate the perimeter

Given length \(l=(x + 6)\) m and width \(w = 7\) m. Substitute into the perimeter formula: \(P=2\times((x + 6)+7)\). Simplify the expression inside the parentheses: \((x + 6)+7=x+13\). Then \(P = 2(x + 13)=2x+26\) meters.

Step3: Calculate the area

Substitute length and width into the area formula: \(A=(x + 6)\times7\). Using the distributive property, \(A = 7x+42\) square meters.

Answer:

The product of \((3x + 6)(3x - 6)\) is \(9x^{2}-36\).

Question 4: