Question

Match the polynomial standard form to its factored form.
1. (1/2 x - 1/4)(4x^2 - 2x + 6)
2. 4(x - 7)
3. (x - 6)(4x - 1)
4. (x + 1)(x + 1)
5. 4x^2(x - 8)
6. x(4x - 3)
7. -2x^2(x^3 - 3x - 4)
8. x(3x^2 + x + 3)

Answer 1

What should I put for the first one then?

Answer 2

To match the polynomial standard forms to their factored forms given in the list, we need to expand each factored form and see which corresponds with the respective polynomial.

Let's go through them one by one:

1. **(1/2 x - 1/4)(4x^2 - 2x + 6)**:
- Expanding gives:
\[
\frac{1}{2}x(4x^2 - 2x + 6) - \frac{1}{4}(4x^2 - 2x + 6)
= 2x^3 - x^2 + 3x - 1 - (x^2 - \frac{1}{2}x + \frac{3}{4})
= 2x^3 - 2x^2 + \frac{7}{2}x - \frac{3}{4}
\]
- This expands into a polynomial.

2. **4(x - 7)**:
- The expanded form is:
\[
4x - 28
\]

3. **(x - 6)(4x - 1)**:
- The expanded form is:
\[
4x^2 - 1x - 24
\]

4. **(x + 1)(x + 1)**:
- The expanded form is:
\[
x^2 + 2x + 1
\]

5. **4x^2(x - 8)**:
- The expanded form is:
\[
4x^3 - 32x^2
\]

6. **x(4x - 3)**:
- The expanded form is:
\[
4x^2 - 3x
\]

7. **-2x^2(x^3 - 3x - 4)**:
- The expanded form is:
\[
-2x^5 + 6x^3 + 8x^2
\]

8. **x(3x^2 + x + 3)**:
- The expanded form is:
\[
3x^3 + x^2 + 3x
\]

Now we will list the factored forms alongside the standard forms:

1. \( \frac{1}{2}x(4x^2 - 2x + 6) - \frac{1}{4}(4x^2 - 2x + 6) \) → Complex polynomial
2. \( 4(x - 7) \) → \( 4x - 28 \)
3. \( (x - 6)(4x - 1) \) → \( 4x^2 - 1x + 24 \)
4. \( (x + 1)(x + 1) \) → \( x^2 + 2x + 1 \)
5. \( 4x^2(x - 8) \) → \( 4x^3 - 32x^2 \)
6. \( x(4x - 3) \) → \( 4x^2 - 3x \)
7. \( -2x^2(x^3 - 3x - 4) \) → \( -2x^5 + 6x^3 + 8x^2 \)
8. \( x(3x^2 + x + 3) \) → \( 3x^3 + x^2 + 3x \)

Now you can match the expanded forms to their original and factored forms accordingly.