Answer :
To determine which expressions are equivalent to [tex]\(57f - 19f^3\)[/tex], we need to expand each of the provided expressions and see if they match the original expression.
Let's go through each option one by one:
Option A: [tex]\(-19f(3 + f^2)\)[/tex]
- Expand this expression:
[tex]\(-19f \times 3 + (-19f \times f^2) = -57f - 19f^3\)[/tex]
This does not match [tex]\(57f - 19f^3\)[/tex], so Option A is not equivalent.
Option B: [tex]\(19(3f^2 - f^2)\)[/tex]
- Expand this expression:
[tex]\(19 \times 3f^2 - 19 \times f^2 = 57f^2 - 19f^2\)[/tex]
This is not equivalent to [tex]\(57f - 19f^3\)[/tex] because it involves [tex]\(f^2\)[/tex] instead of just [tex]\(f\)[/tex] and [tex]\(f^3\)[/tex]. Therefore, Option B is not equivalent.
Option C: [tex]\(19f(3 - f^2)\)[/tex]
- Expand this expression:
[tex]\(19f \times 3 - 19f \times f^2 = 57f - 19f^3\)[/tex]
This matches the original expression perfectly. Thus, Option C is equivalent.
Option D: [tex]\(-19f(-3 + f^2)\)[/tex]
- Expand this expression:
[tex]\((-19f) \times (-3) + (-19f) \times f^2 = 57f - 19f^3\)[/tex]
This matches the original expression exactly. So, Option D is equivalent.
Option E: [tex]\(19(3f - f^3)\)[/tex]
- Expand this expression:
[tex]\(19 \times 3f - 19 \times f^3 = 57f - 19f^3\)[/tex]
This also perfectly matches the original expression. Hence, Option E is equivalent.
Therefore, the correct expressions equivalent to [tex]\(57f - 19f^3\)[/tex] are:
- Option C: [tex]\(19f(3 - f^2)\)[/tex]
- Option D: [tex]\(-19f(-3 + f^2)\)[/tex]
- Option E: [tex]\(19(3f - f^3)\)[/tex]
These are the expressions that are equivalent to the original expression.
Let's go through each option one by one:
Option A: [tex]\(-19f(3 + f^2)\)[/tex]
- Expand this expression:
[tex]\(-19f \times 3 + (-19f \times f^2) = -57f - 19f^3\)[/tex]
This does not match [tex]\(57f - 19f^3\)[/tex], so Option A is not equivalent.
Option B: [tex]\(19(3f^2 - f^2)\)[/tex]
- Expand this expression:
[tex]\(19 \times 3f^2 - 19 \times f^2 = 57f^2 - 19f^2\)[/tex]
This is not equivalent to [tex]\(57f - 19f^3\)[/tex] because it involves [tex]\(f^2\)[/tex] instead of just [tex]\(f\)[/tex] and [tex]\(f^3\)[/tex]. Therefore, Option B is not equivalent.
Option C: [tex]\(19f(3 - f^2)\)[/tex]
- Expand this expression:
[tex]\(19f \times 3 - 19f \times f^2 = 57f - 19f^3\)[/tex]
This matches the original expression perfectly. Thus, Option C is equivalent.
Option D: [tex]\(-19f(-3 + f^2)\)[/tex]
- Expand this expression:
[tex]\((-19f) \times (-3) + (-19f) \times f^2 = 57f - 19f^3\)[/tex]
This matches the original expression exactly. So, Option D is equivalent.
Option E: [tex]\(19(3f - f^3)\)[/tex]
- Expand this expression:
[tex]\(19 \times 3f - 19 \times f^3 = 57f - 19f^3\)[/tex]
This also perfectly matches the original expression. Hence, Option E is equivalent.
Therefore, the correct expressions equivalent to [tex]\(57f - 19f^3\)[/tex] are:
- Option C: [tex]\(19f(3 - f^2)\)[/tex]
- Option D: [tex]\(-19f(-3 + f^2)\)[/tex]
- Option E: [tex]\(19(3f - f^3)\)[/tex]
These are the expressions that are equivalent to the original expression.
