Question

Use either a tabular model or the distributive property to mutiply (x-5)(6x^2-4x+3

Answer 1

To multiply \((x - 5)(6x^2 - 4x + 3)\) using the distributive property (often referred to as the FOIL method for binomials), we will distribute each term in the first polynomial to each term in the second polynomial.

Here’s how it’s done step-by-step:

1. **Distribute \(x\) to each term in \(6x^2 - 4x + 3\):**
- \(x \cdot 6x^2 = 6x^3\)
- \(x \cdot (-4x) = -4x^2\)
- \(x \cdot 3 = 3x\)

So, from the first term \(x\), we get:
\[
6x^3 - 4x^2 + 3x
\]

2. **Distribute \(-5\) to each term in \(6x^2 - 4x + 3\):**
- \(-5 \cdot 6x^2 = -30x^2\)
- \(-5 \cdot (-4x) = 20x\)
- \(-5 \cdot 3 = -15\)

So, from the second term \(-5\), we get:
\[
-30x^2 + 20x - 15
\]

3. **Combine all the results:**
Now we combine the results from both distributions:
\[
6x^3 + (-4x^2) + 3x + (-30x^2) + 20x - 15
\]

Combining like terms:
- The \(x^2\) terms: \(-4x^2 - 30x^2 = -34x^2\)
- The \(x\) terms: \(3x + 20x = 23x\)

4. **Final result:**
Combine all the terms together:
\[
6x^3 - 34x^2 + 23x - 15
\]

Thus, the product of \((x - 5)(6x^2 - 4x + 3)\) is:

\[
\boxed{6x^3 - 34x^2 + 23x - 15}
\]