Answer :
To solve the problem of finding which expressions are equivalent to [tex]\((-7x^3 + 9x^2 - 3) \cdot (-2x^2 - 5x + 6)\)[/tex], we'll follow these steps:
### Step 1: Understand the Expression
We're given the expression [tex]\((-7x^3 + 9x^2 - 3) \cdot (-2x^2 - 5x + 6)\)[/tex]. This expression represents the multiplication of two polynomials.
### Step 2: Expand the Expression
To find the equivalent expression, we expand the product of these two polynomials. Here's how this works:
1. Distribute: Multiply each term in the first polynomial by each term in the second polynomial.
- [tex]\((-7x^3) \cdot (-2x^2) = 14x^5\)[/tex]
- [tex]\((-7x^3) \cdot (-5x) = 35x^4\)[/tex]
- [tex]\((-7x^3) \cdot 6 = -42x^3\)[/tex]
- [tex]\((9x^2) \cdot (-2x^2) = -18x^4\)[/tex]
- [tex]\((9x^2) \cdot (-5x) = -45x^3\)[/tex]
- [tex]\((9x^2) \cdot 6 = 54x^2\)[/tex]
- [tex]\((-3) \cdot (-2x^2) = 6x^2\)[/tex]
- [tex]\((-3) \cdot (-5x) = 15x\)[/tex]
- [tex]\((-3) \cdot 6 = -18\)[/tex]
2. Combine Like Terms: Group terms with the same degree.
- [tex]\(14x^5\)[/tex]
- [tex]\((35x^4 - 18x^4) = 17x^4\)[/tex]
- [tex]\((-42x^3 - 45x^3) = -87x^3\)[/tex]
- [tex]\((54x^2 + 6x^2) = 60x^2\)[/tex]
- [tex]\(15x\)[/tex]
- [tex]\(-18\)[/tex]
3. This gives us the expanded expression: [tex]\(14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18\)[/tex].
### Step 3: Identify Equivalent Expressions
Compare the expanded expression [tex]\(14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18\)[/tex] to the given options:
- Option 1: [tex]\(14x^5 - 17x^4 - 87x^3 + 60x^2 + 15x - 18\)[/tex] (Not equivalent, since it has [tex]\(-17x^4\)[/tex] instead of [tex]\(17x^4\)[/tex])
- Option 2: [tex]\(14x^5 + 35x^4 - 18x^4 - 87x^3 + 60x^2 + 15x - 18\)[/tex] (Equivalent after combining like terms: [tex]\(35x^4 - 18x^4 = 17x^4\)[/tex])
- Option 3: [tex]\(14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18\)[/tex] (Equivalent, already matches expanded form)
- Option 4: [tex]\(14x^5 + 17x^4 - 87x^3 + 40x^2 + 20x^2 + 15x - 18\)[/tex] (Equivalent after combining like terms: [tex]\(40x^2 + 20x^2 = 60x^2\)[/tex])
- Option 5: [tex]\(14x^5 + 53x^4 - 87x^3 + 60x^2 + 15x - 18\)[/tex] (Not equivalent, it has [tex]\(53x^4\)[/tex] instead of [tex]\(17x^4\)[/tex])
Thus, the equivalent expressions are from options 2, 3, and 4.
### Step 1: Understand the Expression
We're given the expression [tex]\((-7x^3 + 9x^2 - 3) \cdot (-2x^2 - 5x + 6)\)[/tex]. This expression represents the multiplication of two polynomials.
### Step 2: Expand the Expression
To find the equivalent expression, we expand the product of these two polynomials. Here's how this works:
1. Distribute: Multiply each term in the first polynomial by each term in the second polynomial.
- [tex]\((-7x^3) \cdot (-2x^2) = 14x^5\)[/tex]
- [tex]\((-7x^3) \cdot (-5x) = 35x^4\)[/tex]
- [tex]\((-7x^3) \cdot 6 = -42x^3\)[/tex]
- [tex]\((9x^2) \cdot (-2x^2) = -18x^4\)[/tex]
- [tex]\((9x^2) \cdot (-5x) = -45x^3\)[/tex]
- [tex]\((9x^2) \cdot 6 = 54x^2\)[/tex]
- [tex]\((-3) \cdot (-2x^2) = 6x^2\)[/tex]
- [tex]\((-3) \cdot (-5x) = 15x\)[/tex]
- [tex]\((-3) \cdot 6 = -18\)[/tex]
2. Combine Like Terms: Group terms with the same degree.
- [tex]\(14x^5\)[/tex]
- [tex]\((35x^4 - 18x^4) = 17x^4\)[/tex]
- [tex]\((-42x^3 - 45x^3) = -87x^3\)[/tex]
- [tex]\((54x^2 + 6x^2) = 60x^2\)[/tex]
- [tex]\(15x\)[/tex]
- [tex]\(-18\)[/tex]
3. This gives us the expanded expression: [tex]\(14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18\)[/tex].
### Step 3: Identify Equivalent Expressions
Compare the expanded expression [tex]\(14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18\)[/tex] to the given options:
- Option 1: [tex]\(14x^5 - 17x^4 - 87x^3 + 60x^2 + 15x - 18\)[/tex] (Not equivalent, since it has [tex]\(-17x^4\)[/tex] instead of [tex]\(17x^4\)[/tex])
- Option 2: [tex]\(14x^5 + 35x^4 - 18x^4 - 87x^3 + 60x^2 + 15x - 18\)[/tex] (Equivalent after combining like terms: [tex]\(35x^4 - 18x^4 = 17x^4\)[/tex])
- Option 3: [tex]\(14x^5 + 17x^4 - 87x^3 + 60x^2 + 15x - 18\)[/tex] (Equivalent, already matches expanded form)
- Option 4: [tex]\(14x^5 + 17x^4 - 87x^3 + 40x^2 + 20x^2 + 15x - 18\)[/tex] (Equivalent after combining like terms: [tex]\(40x^2 + 20x^2 = 60x^2\)[/tex])
- Option 5: [tex]\(14x^5 + 53x^4 - 87x^3 + 60x^2 + 15x - 18\)[/tex] (Not equivalent, it has [tex]\(53x^4\)[/tex] instead of [tex]\(17x^4\)[/tex])
Thus, the equivalent expressions are from options 2, 3, and 4.
