Which are equivalent expressions for (4−2z2−z)(3z+5)

Question

? 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

Answer 1

To find equivalent expressions for \((4−2z^2−z)(3z+5)\), we need to first expand this expression.

1. Distributing \( (4 - 2z^2 - z) \) over \( (3z + 5) \):

\[ (4)(3z) + (4)(5) + (-2z^2)(3z) + (-2z^2)(5) + (-z)(3z) + (-z)(5) \]

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 let's combine all these results:

• 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: \( 20 \)

Putting this all together, we get:

\[ -6z^3 - 13z^2 + 7z + 20 \]

Now let's evaluate the responses given:

1. \(6z^3−10z^2+5z+12z+20−3z^2\)

   • Simplifying this:
   • \( 6z^3 - 10z^2 + (5z + 12z) + 20 - 3z^2 = 6z^3 - 10z^2 + 17z + 20\) which is not equivalent.

2. \(-6z^3−13z^2+7z+20\)

   • This is exactly the expanded form we obtained, so it is equivalent.

3. \(-6z^3−7z^2−2z+20\)

   • This is not equivalent, since the coefficients of \( z^2 \) and \( z \) do not match.

4. \(-6z^3−10z^2−3z^2−5z+12z+20\)

   • Simplifying:
   • \( -6z^3 + (-10z^2 - 3z^2) + (12z - 5z) + 20 = -6z^3 - 13z^2 + 7z + 20\). This is again equivalent.

5. \(-6z^3−13z^2−5z+12z+20\)

   • Simplifying:
   • \( -6z^3 - 13z^2 + (12z - 5z) + 20 = -6z^3 - 13z^2 + 7z + 20\), which is also equivalent.

Thus, the equivalent expressions are:

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