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To determine which of the given products result in either a difference of squares or a perfect square trinomial, let's analyze each expression step-by-step.
### 1. [tex]\((5x+3)(5x-3)\)[/tex]
In this product:
- The first term is [tex]\(5x\)[/tex] in both factors.
- The second terms are [tex]\(+3\)[/tex] and [tex]\(-3\)[/tex], which are opposites.
This fits the form of a difference of squares: [tex]\(a^2 - b^2\)[/tex]. Therefore, [tex]\((5x+3)(5x-3) = (5x)^2 - (3)^2 = 25x^2 - 9\)[/tex], which is a difference of squares.
Result: This product is a difference of squares.
### 2. [tex]\((7x+4)(7x+4)\)[/tex]
This is simply the square of a binomial:
- Both factors are identical [tex]\((7x + 4)\)[/tex].
This fits the form [tex]\((a+b)^2\)[/tex], which is a perfect square trinomial. [tex]\((7x+4)^2 = (7x)^2 + 2(7x)(4) + 4^2 = 49x^2 + 56x + 16\)[/tex].
Result: This product is a perfect square trinomial.
### 3. [tex]\((2x+1)(x+2)\)[/tex]
For this product:
- First factor is [tex]\(2x + 1\)[/tex].
- Second factor is [tex]\(x + 2\)[/tex].
They are neither identical nor are their second terms additive inverses of each other. Therefore, this expression does not fit the criteria for either a difference of squares or a perfect square trinomial.
Result: This product is neither.
### 4. [tex]\((4x-6)(x+8)\)[/tex]
For this product:
- First factor is [tex]\(4x - 6\)[/tex].
- Second factor is [tex]\(x + 8\)[/tex].
Like the previous case, they are neither identical nor are their second terms additive inverses of each other. Thus, this expression does not satisfy the conditions for either a difference of squares or a perfect square trinomial.
Result: This product is neither.
### 5. [tex]\((x-9)(x-9)\)[/tex]
This is another case of a binomial squared:
- Both factors are identical [tex]\((x - 9)\)[/tex].
This expression fits the form [tex]\((a-b)^2\)[/tex]. [tex]\((x-9)^2 = x^2 - 2(9)x + 9^2 = x^2 - 18x + 81\)[/tex], which is a perfect square trinomial.
Result: This product is a perfect square trinomial.
### 6. [tex]\((-3x-6)(-3x+6)\)[/tex]
In this product:
- The first term is [tex]\(-3x\)[/tex] in both factors.
- The second terms are [tex]\(-6\)[/tex] and [tex]\(+6\)[/tex], which are opposites.
This fits the form of a difference of squares: [tex]\(a^2 - b^2\)[/tex]. Therefore, [tex]\((-3x-6)(-3x+6) = (-3x)^2 - (6)^2 = 9x^2 - 36\)[/tex], which is a difference of squares.
Result: This product is a difference of squares.
### Summary
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]
Thus, the answers are:
[tex]\((5 x+3)(5 x-3)\)[/tex]
[tex]\((7 x+4)(7 x+4)\)[/tex]
[tex]\((x-9)(x-9)\)[/tex]
[tex]\((-3 x-6)(-3 x+6)\)[/tex]
### 1. [tex]\((5x+3)(5x-3)\)[/tex]
In this product:
- The first term is [tex]\(5x\)[/tex] in both factors.
- The second terms are [tex]\(+3\)[/tex] and [tex]\(-3\)[/tex], which are opposites.
This fits the form of a difference of squares: [tex]\(a^2 - b^2\)[/tex]. Therefore, [tex]\((5x+3)(5x-3) = (5x)^2 - (3)^2 = 25x^2 - 9\)[/tex], which is a difference of squares.
Result: This product is a difference of squares.
### 2. [tex]\((7x+4)(7x+4)\)[/tex]
This is simply the square of a binomial:
- Both factors are identical [tex]\((7x + 4)\)[/tex].
This fits the form [tex]\((a+b)^2\)[/tex], which is a perfect square trinomial. [tex]\((7x+4)^2 = (7x)^2 + 2(7x)(4) + 4^2 = 49x^2 + 56x + 16\)[/tex].
Result: This product is a perfect square trinomial.
### 3. [tex]\((2x+1)(x+2)\)[/tex]
For this product:
- First factor is [tex]\(2x + 1\)[/tex].
- Second factor is [tex]\(x + 2\)[/tex].
They are neither identical nor are their second terms additive inverses of each other. Therefore, this expression does not fit the criteria for either a difference of squares or a perfect square trinomial.
Result: This product is neither.
### 4. [tex]\((4x-6)(x+8)\)[/tex]
For this product:
- First factor is [tex]\(4x - 6\)[/tex].
- Second factor is [tex]\(x + 8\)[/tex].
Like the previous case, they are neither identical nor are their second terms additive inverses of each other. Thus, this expression does not satisfy the conditions for either a difference of squares or a perfect square trinomial.
Result: This product is neither.
### 5. [tex]\((x-9)(x-9)\)[/tex]
This is another case of a binomial squared:
- Both factors are identical [tex]\((x - 9)\)[/tex].
This expression fits the form [tex]\((a-b)^2\)[/tex]. [tex]\((x-9)^2 = x^2 - 2(9)x + 9^2 = x^2 - 18x + 81\)[/tex], which is a perfect square trinomial.
Result: This product is a perfect square trinomial.
### 6. [tex]\((-3x-6)(-3x+6)\)[/tex]
In this product:
- The first term is [tex]\(-3x\)[/tex] in both factors.
- The second terms are [tex]\(-6\)[/tex] and [tex]\(+6\)[/tex], which are opposites.
This fits the form of a difference of squares: [tex]\(a^2 - b^2\)[/tex]. Therefore, [tex]\((-3x-6)(-3x+6) = (-3x)^2 - (6)^2 = 9x^2 - 36\)[/tex], which is a difference of squares.
Result: This product is a difference of squares.
### Summary
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]
Thus, the answers are:
[tex]\((5 x+3)(5 x-3)\)[/tex]
[tex]\((7 x+4)(7 x+4)\)[/tex]
[tex]\((x-9)(x-9)\)[/tex]
[tex]\((-3 x-6)(-3 x+6)\)[/tex]
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