Math

QuestionIdentify expressions that can be factored as (ax+b)(axb)(a x+b)(a x-b) with integers aa and bb. Options: A x29x^{2}-9, B x211x^{2}-11, C 4x214 x^{2}-1, D 4x224 x^{2}-2, E 9x249 x^{2}-4.

Studdy Solution

STEP 1

Assumptions1. The expressions are in the form (ax+b)(axb)(a x+b)(a x-b), where aa and bb are integers. . We need to find which of the given expressions can be written in this form.

STEP 2

The expression (ax+b)(axb)(a x+b)(a x-b) can be expanded to a2x2b2a^2 x^2 - b^2. This is a difference of squares, which means we are looking for expressions in the form of a2x2b2a^2 x^2 - b^2.

STEP 3

Let's start with option A x29x^{2}-9. We can rewrite this as 12x2321^2 x^2 -3^2. Here, a=1a=1 and b=3b=3, both of which are integers. So, option A can be written in the form (ax+b)(axb)(a x+b)(a x-b).

STEP 4

Next, consider option B x211x^{2}-11. This cannot be written in the form a2x2b2a^2 x^2 - b^2 where aa and bb are integers, because11 is not a perfect square. So, option B cannot be written in the form (ax+b)(axb)(a x+b)(a x-b).

STEP 5

Now, consider option C 4x214 x^{2}-1. We can rewrite this as 22x2122^2 x^2 -1^2. Here, a=2a=2 and b=1b=1, both of which are integers. So, option C can be written in the form (ax+b)(axb)(a x+b)(a x-b).

STEP 6

Next, consider option D 4x224 x^{2}-2. This cannot be written in the form a2x2b2a^2 x^2 - b^2 where aa and bb are integers, because2 is not a perfect square. So, option D cannot be written in the form (ax+b)(axb)(a x+b)(a x-b).

STEP 7

Finally, consider option 9x249 x^{2}-4. We can rewrite this as 32x2223^2 x^2 -2^2. Here, a=3a=3 and b=2b=2, both of which are integers. So, option can be written in the form (ax+b)(axb)(a x+b)(a x-b).
So, the expressions that can be written as (ax+b)(axb)(a x+b)(a x-b), where aa and bb are integers, are options A, C, and.

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