To determine which expressions are equivalent to \((4 - 2z^2 - z)(3z + 5)\), we first need to expand the expression:
\[ (4 - 2z^2 - z)(3z + 5) \]
We can use the distributive property (FOIL method) to expand this:
\[ = 4(3z) + 4(5) - 2z^2(3z) - 2z^2(5) - z(3z) - z(5) \] \[ = 12z + 20 - 6z^3 - 10z^2 - 3z^2 - 5z \] Now, combine like terms:
\[ = -6z^3 + (12z - 5z) + (20) + (-10z^2 - 3z^2) \] \[ = -6z^3 - 13z^2 + 7z + 20 \]
Now, let’s evaluate the provided expressions for equivalency:
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\(6z^3 - 10z^2 + 5z + 12z + 20 - 3z^2\): This simplifies to \(6z^3 - 13z^2 + 17z + 20\). Not equivalent.
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\(-6z^3 - 13z^2 + 7z + 20\): This expression matches our derived expression. Equivalent.
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\(-6z^3 - 7z^2 - 2z + 20\): This does not match our derived expression. Not equivalent.
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\(-6z^3 - 10z^2 - 3z^2 - 5z + 12z + 20\): This simplifies to \(-6z^3 - 13z^2 + 7z + 20\). Equivalent.
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\(-6z^3 - 13z^2 - 5z + 12z + 20\): This simplifies to \(-6z^3 - 13z^2 + 7z + 20\). Equivalent.
The equivalent expressions are:
- \(-6z^3 - 13z^2 + 7z + 20\)
- \(-6z^3 - 10z^2 - 3z^2 - 5z + 12z + 20\)
- \(-6z^3 - 13z^2 - 5z + 12z + 20\)