Question

factoriser
\\(a = (x - 1)^2 - (4x - 2)^2\\)
\\(b = 9x^2 - (x + 1)^2\\)
\\(c = (2x + 3)^2 - (1 - x)^2\\)
\\(d = (3x - 2)^2 - (5x + 1)^2\\)
\\(e = x^2 - 6x + 9 - (x + 3)(x - 2)\\)
\\(f = 25 - (2x + 3)^2\\)
\\(g = 3x^2 - 6x + 3\\)
\\(h = (3x + 3) - (x + 1)(2x - 1)\\)

factoriser en utilisant au préalable une identité remarquable.
\\(a = x^2 - 4 + (x + 2)(x + 3)\\)
\\(b = x^2 + 6x + 9 - (x + 3)(x - 1)\\)
\\(c = (3x - 2)(x + 5) + 9x^2 - 4\\)
\\(d = 9x^2 - 1 + (3x + 1)(2x + 3)\\)
\\(e = x^2 - 4x + 4 + (x + 3)(x - 2)\\)

\\(a = \frac{1}{4} - 25x^2\\)
\\(b = \frac{x^2}{36} - \frac{25}{49}\\)
\\(c = \frac{4}{9}x^2 + \frac{49}{36} + \frac{14}{9}x\\)
\\(d = \frac{81}{16}x^2 - \frac{33}{2}x + \frac{121}{9}\\)
\\(e = \frac{25}{4}x^2 - \frac{169}{144}\\)

Explanation:

This problem asks us to factorize five algebraic expressions labeled \(A\) through \(E\) under the instruction "Factoriser en utilisant au préalable une identité remarquable" (Factorize by first using a remarkable identity / algebraic identity).

Let's analyze and factorize each expression step-by-step.

Factorize Expression A

We are given:

$$A = x^2 - 4 + (x + 2)(x + 3)$$

First, recognize the difference of squares identity:

$$x^2 - 4 = (x - 2)(x + 2)$$

Substitute this back into the expression:

$$A = (x - 2)(x + 2) + (x + 2)(x + 3)$$

Now, extract the common factor \((x + 2)\):

$$A = (x + 2)[(x - 2) + (x + 3)]$$

Simplify the expression inside the brackets:

$$A = (x + 2)(2x + 1)$$

Factorize Expression B

We are given:

$$B = x^2 + 6x + 9 - (x + 3)(x - 1)$$

First, recognize the perfect square trinomial identity:

$$x^2 + 6x + 9 = (x + 3)^2$$

Substitute this back into the expression:

$$B = (x + 3)^2 - (x + 3)(x - 1)$$

Now, extract the common factor \((x + 3)\):

$$B = (x + 3)[(x + 3) - (x - 1)]$$

Simplify the expression inside the brackets:

$$B = (x + 3)(x + 3 - x + 1) = (x + 3)(4)$$

Thus, the factorized form is:

$$B = 4(x + 3)$$

Factorize Expression C

We are given:

$$C = (3x - 2)(x + 5) + 9x^2 - 4$$

First, recognize the difference of squares identity:

$$9x^2 - 4 = (3x)^2 - 2^2 = (3x - 2)(3x + 2)$$

Substitute this back into the expression:

$$C = (3x - 2)(x + 5) + (3x - 2)(3x + 2)$$

Now, extract the common factor \((3x - 2)\):

$$C = (3x - 2)[(x + 5) + (3x + 2)]$$

Simplify the expression inside the brackets:

$$C = (3x - 2)(4x + 7)$$

Factorize Expression D

We are given:

$$D = 9x^2 - 1 + (3x + 1)(2x + 3)$$

First, recognize the difference of squares identity:

$$9x^2 - 1 = (3x)^2 - 1^2 = (3x - 1)(3x + 1)$$

Substitute this back into the expression:

$$D = (3x - 1)(3x + 1) + (3x + 1)(2x + 3)$$

Now, extract the common factor \((3x + 1)\):

$$D = (3x + 1)[(3x - 1) + (2x + 3)]$$

Simplify the expression inside the brackets:

$$D = (3x + 1)(5x + 2)$$

Factorize Expression E

We are given:

$$E = x^2 - 4x + 4 + (x + 3)(x - 2)$$

First, recognize the perfect square trinomial identity:

$$x^2 - 4x + 4 = (x - 2)^2$$

Substitute this back into the expression:

$$E = (x - 2)^2 + (x + 3)(x - 2)$$

Now, extract the common factor \((x - 2)\):

$$E = (x - 2)[(x - 2) + (x + 3)]$$

Simplify the expression inside the brackets:

$$E = (x - 2)(2x + 1)$$

Answer:

Question 1

$$A = (x + 2)(2x + 1)$$

Question 2

$$B = 4(x + 3)$$

Question 3

$$C = (3x - 2)(4x + 7)$$

Question 4

$$D = (3x + 1)(5x + 2)$$

Question 5

$$E = (x - 2)(2x + 1)$$