Question

Question

Which are equivalent expressions for (4−2z2−z)(3z+5)
? Select all that apply.(3 points)
Responses

6z3−10z2+5z+12z+20−3z2
6 z cubed minus 10 z squared plus 5 z plus 12 z plus 20 minus 3 z squared

−6z3−13z2+7z+20
negative 6 z cubed minus 13 z squared plus 7 z plus 20

−6z3−7z2−2z+20
negative 6 z cubed minus 7 z squared minus 2 z plus 20

−6z3−10z2−3z2−5z+12z+20
negative 6 z cubed minus 10 z squared minus 3 z squared minus 5 z plus 12 z plus 20

−6z3−13z2−5z+12z+20
negative 6 z cubed minus 13 z squared minus 5 z plus 12 z plus 20
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Answer 1

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:

1. \(6z^3 - 10z^2 + 5z + 12z + 20 - 3z^2\): This simplifies to \(6z^3 - 13z^2 + 17z + 20\). Not equivalent.

2. \(-6z^3 - 13z^2 + 7z + 20\): This expression matches our derived expression. Equivalent.

3. \(-6z^3 - 7z^2 - 2z + 20\): This does not match our derived expression. Not equivalent.

4. \(-6z^3 - 10z^2 - 3z^2 - 5z + 12z + 20\): This simplifies to \(-6z^3 - 13z^2 + 7z + 20\). Equivalent.

5. \(-6z^3 - 13z^2 - 5z + 12z + 20\): This simplifies to \(-6z^3 - 13z^2 + 7z + 20\). Equivalent.

The equivalent expressions are:

• \(-6z^3 - 13z^2 + 7z + 20\)
• \(-6z^3 - 10z^2 - 3z^2 - 5z + 12z + 20\)
• \(-6z^3 - 13z^2 - 5z + 12z + 20\)