Sure, let's break down the steps Lorne used to subtract the polynomial [tex]\(6x^3 - 2x + 3\)[/tex] from [tex]\(-3x^3 + 5x^2 + 4x - 7\)[/tex].
1. Initial setup:
[tex]\[
\left(-3x^3 + 5x^2 + 4x - 7\right) + \left(-6x^3 + 2x - 3\right)
\][/tex]
Here, Lorne is setting up the problem by adding the opposite of the polynomial that needs to be subtracted.
2. Distributing within the polynomials:
[tex]\[
\left(-3x^3\right) + 5x^2 + 4x + (-7) + \left(-6x^3\right) + 2x + (-3)
\][/tex]
Lorne writes out each term of both polynomials separately so that they can be grouped by similar terms.
3. Grouping like terms:
[tex]\[
\left[\left(-3x^3\right) + \left(-6x^3\right)\right] + [4x + 2x] + [(-7) + (-3)] + \left[5x^2\right]
\][/tex]
Now, the like terms (terms with the same degree) are grouped together for simplification.
4. Simplifying each group of like terms:
[tex]\[
-9x^3 + 6x + (-10) + 5x^2
\][/tex]
Lorne adds the coefficients of the terms with the same degrees:
- [tex]\((-3x^3) + (-6x^3) = -9x^3\)[/tex]
- [tex]\(4x + 2x = 6x\)[/tex]
- [tex]\((-7) + (-3) = -10\)[/tex]
- [tex]\(5x^2\)[/tex] remains as is because there is no similar term to combine with.
5. Writing the polynomial in standard form:
[tex]\[
-9x^3 + 5x^2 + 6x - 10
\][/tex]
All of the simplified terms are combined into the final polynomial, ordered by the decreasing power of [tex]\(x\)[/tex].
These steps clearly show the process Lorne used to find the difference between the two polynomials.
