To find equivalent expressions for \((4−2z^2−z)(3z+5)\), we need to first expand the expression.
Using the distributive property (also known as the FOIL method for binomials), we can expand as follows:
\[ (4 - 2z^2 - z)(3z + 5) = (4)(3z) + (4)(5) + (-2z^2)(3z) + (-2z^2)(5) + (-z)(3z) + (-z)(5) \]
Calculating each term:
- \(4 \cdot 3z = 12z\)
- \(4 \cdot 5 = 20\)
- \(-2z^2 \cdot 3z = -6z^3\)
- \(-2z^2 \cdot 5 = -10z^2\)
- \(-z \cdot 3z = -3z^2\)
- \(-z \cdot 5 = -5z\)
Now, combine these terms:
\[ -6z^3 + (20 + 12z - 5z) + (-10z^2 - 3z^2) \]
Combining like terms:
- The \(z^3\) term: \(-6z^3\)
- The \(z^2\) terms: \(-10z^2 - 3z^2 = -13z^2\)
- The \(z\) terms: \((12z - 5z) = 7z\)
- The constant term: \(20\)
So, the expanded form is:
\[ -6z^3 - 13z^2 + 7z + 20 \]
Now, let's compare this expression with the options you've provided:
- \(6z^3 - 10z^2 + 5z + 12z + 20 - 3z^2\): This simplifies to \(6z^3 - 13z^2 + 20\) which is not equivalent.
- \(-6z^3 - 13z^2 + 7z + 20\): This is equivalent.
- \(-6z^3 - 7z^2 - 2z + 20\): This is not equivalent.
- \(-6z^3 - 7z^2 - 2z + 20\): Duplicate option or typo; same as above and not equivalent.
- \(-6z^3 - 10z^2 - 3z^2 - 5z + 12z + 20\): This simplifies to \(-6z^3 - 13z^2 + 7z + 20\), which is equivalent.
- \(-6z^3 - 13z^2 - 5z + 12z + 20\): Simplifies to \(-6z^3 - 13z^2 + 7z + 20\) which is equivalent.
Thus, the equivalent expressions are:
- \(-6z^3 - 13z^2 + 7z + 20\)
- \(-6z^3 - 13z^2 + 7z + 20\) (the result from option 5).
- \(-6z^3 - 13z^2 + 7z + 20\) (the result from option 6).
Final selections:
- \(-6z^3 - 13z^2 + 7z + 20\)
- \(-6z^3 - 13z^2 + 7z + 20\) (as repeated).
- \(-6z^3 - 13z^2 + 7z + 20\) (as repeated).
Therefore, the equivalent expressions are all those that simplify down to the same polynomial form: \(-6z^3 - 13z^2 + 7z + 20\).
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