Answer :
To identify which products result in a difference of squares or a perfect square trinomial, we need to know a few algebraic identities:
1. Difference of Squares:
[tex]\((a + b)(a - b) = a^2 - b^2\)[/tex]
2. Perfect Square Trinomial:
[tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex] or [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]
Let's analyze each product:
1. [tex]\((5x + 3)(5x - 3)\)[/tex]
- This follows the form of a difference of squares: [tex]\(a = 5x\)[/tex] and [tex]\(b = 3\)[/tex].
- [tex]\((5x + 3)(5x - 3) = (5x)^2 - 3^2 = 25x^2 - 9\)[/tex]
- This matches the difference of squares identity.
2. [tex]\((7x + 4)(7x + 4)\)[/tex]
- This follows the form of a perfect square trinomial: [tex]\(a = 7x\)[/tex] and [tex]\(b = 4\)[/tex].
- [tex]\((7x + 4)^2 = (7x)^2 + 2(7x)(4) + 4^2 = 49x^2 + 56x + 16\)[/tex]
- This matches the perfect square trinomial identity.
3. [tex]\((2x + 1)(x + 2)\)[/tex]
- This does not follow the form of either a difference of squares or a perfect square trinomial.
- Multiplying it out, [tex]\((2x + 1)(x + 2) = 2x^2 + 4x + x + 2 = 2x^2 + 5x + 2\)[/tex]
- This is neither a difference of squares nor a perfect square trinomial.
4. [tex]\((4x - 6)(x + 8)\)[/tex]
- This does not follow the form of either a difference of squares or a perfect square trinomial.
- Multiplying it out, [tex]\((4x - 6)(x + 8) = 4x^2 + 32x - 6x - 48 = 4x^2 + 26x - 48\)[/tex]
- This is neither a difference of squares nor a perfect square trinomial.
5. [tex]\((x - 9)(x - 9)\)[/tex]
- This follows the form of a perfect square trinomial: [tex]\(a = x\)[/tex] and [tex]\(b = 9\)[/tex].
- [tex]\((x - 9)^2 = x^2 - 2(x)(9) + 9^2 = x^2 - 18x + 81\)[/tex]
- This matches the perfect square trinomial identity.
6. [tex]\((-3x - 6)(-3x + 6)\)[/tex]
- This follows the form of a difference of squares: [tex]\(a = -3x\)[/tex] and [tex]\(b = 6\)[/tex].
- [tex]\((-3x - 6)(-3x + 6) = (-3x)^2 - 6^2 = 9x^2 - 36\)[/tex]
- This matches the difference of squares identity.
Based on the above calculations, the products that result in a difference of squares or a perfect square trinomial are:
- [tex]\((5x + 3)(5x - 3)\)[/tex]
- [tex]\((7x + 4)(7x + 4)\)[/tex]
- [tex]\((x - 9)(x - 9)\)[/tex]
- [tex]\((-3x - 6)(-3x + 6)\)[/tex]
1. Difference of Squares:
[tex]\((a + b)(a - b) = a^2 - b^2\)[/tex]
2. Perfect Square Trinomial:
[tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex] or [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]
Let's analyze each product:
1. [tex]\((5x + 3)(5x - 3)\)[/tex]
- This follows the form of a difference of squares: [tex]\(a = 5x\)[/tex] and [tex]\(b = 3\)[/tex].
- [tex]\((5x + 3)(5x - 3) = (5x)^2 - 3^2 = 25x^2 - 9\)[/tex]
- This matches the difference of squares identity.
2. [tex]\((7x + 4)(7x + 4)\)[/tex]
- This follows the form of a perfect square trinomial: [tex]\(a = 7x\)[/tex] and [tex]\(b = 4\)[/tex].
- [tex]\((7x + 4)^2 = (7x)^2 + 2(7x)(4) + 4^2 = 49x^2 + 56x + 16\)[/tex]
- This matches the perfect square trinomial identity.
3. [tex]\((2x + 1)(x + 2)\)[/tex]
- This does not follow the form of either a difference of squares or a perfect square trinomial.
- Multiplying it out, [tex]\((2x + 1)(x + 2) = 2x^2 + 4x + x + 2 = 2x^2 + 5x + 2\)[/tex]
- This is neither a difference of squares nor a perfect square trinomial.
4. [tex]\((4x - 6)(x + 8)\)[/tex]
- This does not follow the form of either a difference of squares or a perfect square trinomial.
- Multiplying it out, [tex]\((4x - 6)(x + 8) = 4x^2 + 32x - 6x - 48 = 4x^2 + 26x - 48\)[/tex]
- This is neither a difference of squares nor a perfect square trinomial.
5. [tex]\((x - 9)(x - 9)\)[/tex]
- This follows the form of a perfect square trinomial: [tex]\(a = x\)[/tex] and [tex]\(b = 9\)[/tex].
- [tex]\((x - 9)^2 = x^2 - 2(x)(9) + 9^2 = x^2 - 18x + 81\)[/tex]
- This matches the perfect square trinomial identity.
6. [tex]\((-3x - 6)(-3x + 6)\)[/tex]
- This follows the form of a difference of squares: [tex]\(a = -3x\)[/tex] and [tex]\(b = 6\)[/tex].
- [tex]\((-3x - 6)(-3x + 6) = (-3x)^2 - 6^2 = 9x^2 - 36\)[/tex]
- This matches the difference of squares identity.
Based on the above calculations, the products that result in a difference of squares or a perfect square trinomial are:
- [tex]\((5x + 3)(5x - 3)\)[/tex]
- [tex]\((7x + 4)(7x + 4)\)[/tex]
- [tex]\((x - 9)(x - 9)\)[/tex]
- [tex]\((-3x - 6)(-3x + 6)\)[/tex]
