To expand the expression [tex]\((2x + 5)(x^2 - 4x - 1)\)[/tex] into a polynomial in standard form, we will apply the distributive property, also known as the FOIL method (First, Outer, Inner, Last) when dealing with binomials. Here, we will distribute each term in the first polynomial with each term in the second polynomial.
Let's break it down step-by-step:
1. Distribute [tex]\(2x\)[/tex] through the second polynomial [tex]\(x^2 - 4x - 1\)[/tex]:
- [tex]\(2x \cdot x^2 = 2x^3\)[/tex]
- [tex]\(2x \cdot -4x = -8x^2\)[/tex]
- [tex]\(2x \cdot -1 = -2x\)[/tex]
2. Distribute [tex]\(5\)[/tex] through the second polynomial [tex]\(x^2 - 4x - 1\)[/tex]:
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot -4x = -20x\)[/tex]
- [tex]\(5 \cdot -1 = -5\)[/tex]
3. Combine all the terms from the distributions:
[tex]\(2x^3\)[/tex] (from the first set of distributions) \\
[tex]\(-8x^2\)[/tex] (from the first set of distributions) \\
[tex]\(-2x\)[/tex] (from the first set of distributions) \\\
[tex]\(5x^2\)[/tex] (from the second set of distributions) \\
[tex]\(-20x\)[/tex] (from the second set of distributions) \\
[tex]\(-5\)[/tex] (from the second set of distributions)
4. Combine like terms (terms with the same power of [tex]\(x\)[/tex]):
- The cubic term is [tex]\(2x^3\)[/tex].
- For the quadratic terms, combine [tex]\(-8x^2\)[/tex] and [tex]\(5x^2\)[/tex]: [tex]\(-8x^2 + 5x^2 = -3x^2\)[/tex].
- For the linear terms, combine [tex]\(-2x\)[/tex] and [tex]\(-20x\)[/tex]: [tex]\(-2x - 20x = -22x\)[/tex].
- The constant term is [tex]\(-5\)[/tex].
5. Write the final expanded polynomial in standard form:
[tex]\[
2x^3 - 3x^2 - 22x - 5
\][/tex]
Therefore, the expanded expression in polynomial standard form is:
[tex]\[
2x^3 - 3x^2 - 22x - 5
\][/tex]
