Answer :
To determine which products result in a perfect square trinomial, let's analyze each product one by one.
1. Product: [tex]\((-x+9)(-x-9)\)[/tex]
First, distribute the factors:
[tex]\[ (-x+9)(-x-9) = (-x)(-x) + (-x)(-9) + (9)(-x) + (9)(-9) \][/tex]
Simplifying:
[tex]\[ = x^2 + 9x - 9x - 81 = x^2 - 81 \][/tex]
This is not a perfect square trinomial.
2. Product: [tex]\((xy+x)(xy+x)\)[/tex]
Notice this is a square of a binomial:
[tex]\[ (xy+x)(xy+x) = (xy+x)^2 \][/tex]
Applying the square of a binomial formula:
[tex]\[ (a+b)^2 = a^2 + 2ab + b^2 \][/tex]
Let [tex]\(a = xy\)[/tex] and [tex]\(b = x\)[/tex]. Thus:
[tex]\[ (xy+x)^2 = (xy)^2 + 2(xy)(x) + x^2 = x^2y^2 + 2x^2y + x^2 \][/tex]
This is a perfect square trinomial.
3. Product: [tex]\((2x-3)(-3+2x)\)[/tex]
Notice that [tex]\((2x-3)\)[/tex] is the same as [tex]\((-3+2x)\)[/tex]:
[tex]\[ (2x-3)(-3+2x) = (2x-3)(2x-3) = (2x-3)^2 \][/tex]
Using the square of a binomial formula:
[tex]\[ (a-b)^2 = a^2 - 2ab + b^2 \][/tex]
Let [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex]. Thus:
[tex]\[ (2x-3)^2 = (2x)^2 - 2(2x)(3) + 3^2 = 4x^2 - 12x + 9 \][/tex]
This is a perfect square trinomial.
4. Product: [tex]\(\left(16-x^2\right)\left(x^2-16\right)\)[/tex]
Expand the factors:
[tex]\[ (16-x^2)(x^2-16) = 16(x^2) - 16(16) - x^2(x^2) + x^2(16) \][/tex]
Simplifying:
[tex]\[ = 16x^2 - 256 - x^4 + 16x^2 = -x^4 + 32x^2 - 256 \][/tex]
This is not a perfect square trinomial.
5. Product: [tex]\(\left(4y^2+25\right)\left(25+4y^2\right)\)[/tex]
Notice that [tex]\((4y^2 + 25)\)[/tex] is the same as [tex]\((25 + 4y^2)\)[/tex]:
[tex]\[ (4y^2+25)(25+4y^2) = (4y^2 + 25)^2 \][/tex]
However, recognize that:
[tex]\[ (a + b)^2 = a^2 + 2ab + b^2 \][/tex]
Here [tex]\(a = 4y^2\)[/tex] and [tex]\(b = 25\)[/tex], makes:
[tex]\[ (4y^2 + 25)^2 = (4y^2)^2 + 2(4y^2)(25) + (25)^2 = 16y^4 + 200y^2 + 625 \][/tex]
However, this is a perfect square, but not a trinomial form that simplifies directly into a quadratic expression.
Therefore, the products that result in a perfect square trinomial are:
- [tex]\((xy + x)(xy + x)\)[/tex]
- [tex]\((2x - 3)(-3 + 2x)\)[/tex]
These two are perfect square trinomials.
Thus, the options are:
1. [tex]\((xy+x)(xy+x)\)[/tex]
2. [tex]\((2x-3)(-3+2x)\)[/tex]
1. Product: [tex]\((-x+9)(-x-9)\)[/tex]
First, distribute the factors:
[tex]\[ (-x+9)(-x-9) = (-x)(-x) + (-x)(-9) + (9)(-x) + (9)(-9) \][/tex]
Simplifying:
[tex]\[ = x^2 + 9x - 9x - 81 = x^2 - 81 \][/tex]
This is not a perfect square trinomial.
2. Product: [tex]\((xy+x)(xy+x)\)[/tex]
Notice this is a square of a binomial:
[tex]\[ (xy+x)(xy+x) = (xy+x)^2 \][/tex]
Applying the square of a binomial formula:
[tex]\[ (a+b)^2 = a^2 + 2ab + b^2 \][/tex]
Let [tex]\(a = xy\)[/tex] and [tex]\(b = x\)[/tex]. Thus:
[tex]\[ (xy+x)^2 = (xy)^2 + 2(xy)(x) + x^2 = x^2y^2 + 2x^2y + x^2 \][/tex]
This is a perfect square trinomial.
3. Product: [tex]\((2x-3)(-3+2x)\)[/tex]
Notice that [tex]\((2x-3)\)[/tex] is the same as [tex]\((-3+2x)\)[/tex]:
[tex]\[ (2x-3)(-3+2x) = (2x-3)(2x-3) = (2x-3)^2 \][/tex]
Using the square of a binomial formula:
[tex]\[ (a-b)^2 = a^2 - 2ab + b^2 \][/tex]
Let [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex]. Thus:
[tex]\[ (2x-3)^2 = (2x)^2 - 2(2x)(3) + 3^2 = 4x^2 - 12x + 9 \][/tex]
This is a perfect square trinomial.
4. Product: [tex]\(\left(16-x^2\right)\left(x^2-16\right)\)[/tex]
Expand the factors:
[tex]\[ (16-x^2)(x^2-16) = 16(x^2) - 16(16) - x^2(x^2) + x^2(16) \][/tex]
Simplifying:
[tex]\[ = 16x^2 - 256 - x^4 + 16x^2 = -x^4 + 32x^2 - 256 \][/tex]
This is not a perfect square trinomial.
5. Product: [tex]\(\left(4y^2+25\right)\left(25+4y^2\right)\)[/tex]
Notice that [tex]\((4y^2 + 25)\)[/tex] is the same as [tex]\((25 + 4y^2)\)[/tex]:
[tex]\[ (4y^2+25)(25+4y^2) = (4y^2 + 25)^2 \][/tex]
However, recognize that:
[tex]\[ (a + b)^2 = a^2 + 2ab + b^2 \][/tex]
Here [tex]\(a = 4y^2\)[/tex] and [tex]\(b = 25\)[/tex], makes:
[tex]\[ (4y^2 + 25)^2 = (4y^2)^2 + 2(4y^2)(25) + (25)^2 = 16y^4 + 200y^2 + 625 \][/tex]
However, this is a perfect square, but not a trinomial form that simplifies directly into a quadratic expression.
Therefore, the products that result in a perfect square trinomial are:
- [tex]\((xy + x)(xy + x)\)[/tex]
- [tex]\((2x - 3)(-3 + 2x)\)[/tex]
These two are perfect square trinomials.
Thus, the options are:
1. [tex]\((xy+x)(xy+x)\)[/tex]
2. [tex]\((2x-3)(-3+2x)\)[/tex]
