Question

Multiply the following polynomials. Write your answer in standard form.



(
3
x
−
1
)
(
2
x
3
+
4
x
2
−
5
)
=
(3x−1)(2x
3
+4x
2
−5)=

Answer 1

To multiply the polynomials \((3x - 1)(2x^3 + 4x^2 - 5)\), we will use the distributive property (also known as the FOIL method for binomials).

Let's distribute each term in the first polynomial \(3x - 1\) across the second polynomial \(2x^3 + 4x^2 - 5\).

1. **Distributing \(3x\):**
- \(3x \cdot 2x^3 = 6x^4\)
- \(3x \cdot 4x^2 = 12x^3\)
- \(3x \cdot (-5) = -15x\)

2. **Distributing \(-1\):**
- \(-1 \cdot 2x^3 = -2x^3\)
- \(-1 \cdot 4x^2 = -4x^2\)
- \(-1 \cdot (-5) = 5\)

Now we combine all the results:

\[
6x^4 + 12x^3 - 15x - 2x^3 - 4x^2 + 5
\]

Next, we'll combine like terms:

- The \(x^3\) terms: \(12x^3 - 2x^3 = 10x^3\)
- The \(x^2\) term: \(-4x^2\) (only one \(x^2\) term)
- The \(x\) term: \(-15x\) (only one \(x\) term)
- The constant term: \(5\)

Putting it all together in standard form (descending order of the exponent):

\[
6x^4 + 10x^3 - 4x^2 - 15x + 5
\]

Thus, the final answer is:

\[
\boxed{6x^4 + 10x^3 - 4x^2 - 15x + 5}
\]