Question

assessment readiness

1. find the standard form for the product of $(x^4 + 8)$ and $(x^2 - 2)$. choose true or false for each statement about the product.

a. it is a $4^\text{th}$ degree polynomial. $circ$ true $circ$ false
b. the constant term is $-16$. $circ$ true $circ$ false
c. it has 3 terms. $circ$ true $circ$ false

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

a. the coefficient of the $x$-term is $-45$. $circ$ true $circ$ false
b. the leading term is $25x^2$. $circ$ true $circ$ false
c. the constant term is $81$. $circ$ true $circ$ false

1. find the product $(3x + 6)(3x - 6)$. show your work.
2. find the product $(x + 10)(4x + 5)$. show your work.
3. a rectangle has a length $(x + 6)$ m and a width of $7$ m. write expressions to represent the perimeter and area of the rectangle. explain how you determined your answers.

Explanation:

Question 3: Find the product \((3x + 6)(3x - 6)\)

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 we get \(9x^2 - 36\).

Question 4: Find the product \((x + 10)(4x + 5)\)

Step1: Use the distributive property (FOIL method)

Multiply the First terms: \(x\times4x = 4x^2\)
Multiply the Outer terms: \(x\times5 = 5x\)
Multiply the Inner terms: \(10\times4x = 40x\)
Multiply the Last terms: \(10\times5 = 50\)

Step2: Combine like terms

Add the outer and inner terms: \(5x+40x = 45x\)
Combine all terms: \(4x^2+45x + 50\)

Question 5: A rectangle has a length \((x + 6)\) m and a width of 7 m. Write expressions to represent the perimeter and area of the rectangle. Explain how you determined your answers.

For Perimeter:

Step1: Recall the perimeter formula for a rectangle

The formula for the perimeter \(P\) of a rectangle is \(P = 2\times(\text{length}+\text{width})\).

Step2: Substitute the values

Substitute length \(= x + 6\) and width \(= 7\) into the formula:
\(P=2\times((x + 6)+7)\)

Step3: Simplify the expression

First, simplify the expression inside the parentheses: \((x + 6)+7=x + 13\)
Then, multiply by 2: \(2\times(x + 13)=2x+26\)

For Area:

Step1: Recall the area formula for a rectangle

The formula for the area \(A\) of a rectangle is \(A=\text{length}\times\text{width}\).

Step2: Substitute the values

Substitute length \(= x + 6\) and width \(= 7\) into the formula:
\(A=(x + 6)\times7\)

Step3: Simplify the expression

Use the distributive property: \(7\times x+7\times6 = 7x + 42\)

Answer:

\(9x^2 - 36\)