Answer :
To simplify the expression [tex]\((6x-5)(2x^2-3x-6)\)[/tex], we need to use the distributive property, also known as the FOIL method for distributing two binomials. However, in this case, we're distributing a binomial and a trinomial. Let's go through it step-by-step:
1. Distribute [tex]\(6x\)[/tex] to each term in the trinomial [tex]\(2x^2 - 3x - 6\)[/tex]:
- First, multiply [tex]\(6x \times 2x^2\)[/tex] which gives [tex]\(12x^3\)[/tex].
- Then, multiply [tex]\(6x \times -3x\)[/tex] which gives [tex]\(-18x^2\)[/tex].
- Finally, multiply [tex]\(6x \times -6\)[/tex] which gives [tex]\(-36x\)[/tex].
2. Distribute [tex]\(-5\)[/tex] to each term in the trinomial [tex]\(2x^2 - 3x - 6\)[/tex]:
- First, multiply [tex]\(-5 \times 2x^2\)[/tex] which gives [tex]\(-10x^2\)[/tex].
- Then, multiply [tex]\(-5 \times -3x\)[/tex] which gives [tex]\(15x\)[/tex].
- Finally, multiply [tex]\(-5 \times -6\)[/tex] which gives [tex]\(30\)[/tex].
3. Combine all the results:
- The [tex]\(x^3\)[/tex] term is: [tex]\(12x^3\)[/tex].
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-18x^2 - 10x^2 = -28x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-36x + 15x = -21x\)[/tex].
- The constant term is: [tex]\(30\)[/tex].
Putting it all together, the simplified expression is:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
So, the correct simplification of the expression is [tex]\( \boxed{12x^3 - 28x^2 - 21x + 30} \)[/tex].
1. Distribute [tex]\(6x\)[/tex] to each term in the trinomial [tex]\(2x^2 - 3x - 6\)[/tex]:
- First, multiply [tex]\(6x \times 2x^2\)[/tex] which gives [tex]\(12x^3\)[/tex].
- Then, multiply [tex]\(6x \times -3x\)[/tex] which gives [tex]\(-18x^2\)[/tex].
- Finally, multiply [tex]\(6x \times -6\)[/tex] which gives [tex]\(-36x\)[/tex].
2. Distribute [tex]\(-5\)[/tex] to each term in the trinomial [tex]\(2x^2 - 3x - 6\)[/tex]:
- First, multiply [tex]\(-5 \times 2x^2\)[/tex] which gives [tex]\(-10x^2\)[/tex].
- Then, multiply [tex]\(-5 \times -3x\)[/tex] which gives [tex]\(15x\)[/tex].
- Finally, multiply [tex]\(-5 \times -6\)[/tex] which gives [tex]\(30\)[/tex].
3. Combine all the results:
- The [tex]\(x^3\)[/tex] term is: [tex]\(12x^3\)[/tex].
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-18x^2 - 10x^2 = -28x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-36x + 15x = -21x\)[/tex].
- The constant term is: [tex]\(30\)[/tex].
Putting it all together, the simplified expression is:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
So, the correct simplification of the expression is [tex]\( \boxed{12x^3 - 28x^2 - 21x + 30} \)[/tex].
