Expand the expression to a polynomial in standard form:
left parenthesis, 2, x, plus, 3, right parenthesis, left parenthesis, x, squared, minus, 6, x, plus, 1, right parenthesis
(2x+3)(x
2
−6x+1)

To expand the expression \((2x + 3)(x^2 - 6x + 1)\) and write it as a polynomial in standard form, follow these steps:

1. Distribute \(2x\) to each term inside the second parenthesis:
   \[
   2x \cdot (x^2 - 6x + 1) = 2x \cdot x^2 - 2x \cdot 6x + 2x \cdot 1 = 2x^3 - 12x^2 + 2x
   \]

2. Distribute \(3\) to each term inside the second parenthesis:
   \[
   3 \cdot (x^2 - 6x + 1) = 3 \cdot x^2 - 3 \cdot 6x + 3 \cdot 1 = 3x^2 - 18x + 3
   \]

3. Combine the results from steps 1 and 2:
   \[
   (2x^3 - 12x^2 + 2x) + (3x^2 - 18x + 3)
   \]

4. Combine like terms:
   \[
   2x^3 + (-12x^2 + 3x^2) + (2x - 18x) + 3 = 2x^3 - 9x^2 - 16x + 3
   \]

So, the expanded polynomial in standard form is:
\[
2x^3 - 9x^2 - 16x + 3
\]

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you identify the standard form of a polynomial?
2. What are like terms in a polynomial, and why is it important to combine them?
3. How would the expansion process change if one of the terms in the second factor were missing?
4. How do the degrees of the polynomials affect the number of terms in the final expanded form?
5. What happens if you expand a polynomial where the coefficients include fractions?

**Tip:** When multiplying polynomials, always ensure each term from the first polynomial is multiplied by every term in the second polynomial.

Learn how to expand the expression (2x + 3)(x^2 - 6x + 1) into a polynomial in standard form with a step-by-step solution. This problem involves the multiplication of polynomials and standard polynomial notation. Suitable for high school students.

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