Question

1. cedric multiplied (6x + 1) by a second factor to get a product of 36x² − 121. find the second factor.
2. which of the following does not represent a difference of two squares?

a. x⁴ − 9
b. 49x² − 100
c. 25x³ − 4
d. 144x² − 121

1. matthew applied the difference of squares pattern to factor the polynomial below. explain matthew’s error and correct his solution.

4x² − 36
(2x − 6)(2x + 6)

1. lola is catering fajitas for an event. each rectangular tabletop at the event has an area that can be represented by 49x² − 400 units². write expressions to represent the dimensions of each tabletop.

Explanation:

Step1: Problem 5: Identify missing factor

Let the second factor be $A$. We know $(6x + 11)A = 36x^2 - 121$. Notice $36x^2 - 121 = (6x)^2 - 11^2$, which is a difference of squares. The difference of squares formula is $a^2 - b^2 = (a+b)(a-b)$.

Step2: Problem 5: Solve for A

$A = \frac{(6x)^2 - 11^2}{6x + 11} = 6x - 11$

Step3: Problem 6: Define difference of squares

A difference of squares has the form $a^2 - b^2$, where both terms are perfect squares with no odd exponents on variables.

Step4: Problem 6: Analyze each option

• a. $x^4 - 9 = (x^2)^2 - 3^2$ (valid)
• b. $49x^2 - 100 = (7x)^2 - 10^2$ (valid)
• c. $25x^3 - 4$: $x^3$ has an odd exponent, not a perfect square term.
• d. $144x^2 - 121 = (12x)^2 - 11^2$ (valid)

Step5: Problem7: Identify Matthew's error

Matthew factored $4x^2 - 36$ as $(2x-6)(2x+6)$, but he did not factor out the greatest common factor (GCF) first. The GCF of $4x^2$ and $36$ is 4.

Step6: Problem7: Correct the factoring

First factor out the GCF: $4x^2 - 36 = 4(x^2 - 9)$. Then factor the difference of squares: $x^2 - 9 = (x-3)(x+3)$. So the full factorization is $4(x-3)(x+3)$.

Step8: Problem8: Factor the area expression

The area is $49x^2 - 400 = (7x)^2 - 20^2$. Using the difference of squares formula, this factors to $(7x - 20)(7x + 20)$. The dimensions of the rectangle are the two factors.

Answer:

1. $6x - 11$
2. c. $25x^3 - 4$
3. Error: Failed to factor out GCF first. Correct solution: $4(x-3)(x+3)$
4. The dimensions are $7x - 20$ and $7x + 20$