Answer :
To find the product of the polynomials [tex]\((5x^2 - x - 3)(2x + 6)\)[/tex], we can use the distributive property, often referred to as the FOIL method for binomials, but expanded for polynomials. Let's go step by step:
1. Distribute each term of the first polynomial to the second polynomial:
- Multiply [tex]\(5x^2\)[/tex] by each term in [tex]\(2x + 6\)[/tex]:
- [tex]\(5x^2 \cdot 2x = 10x^3\)[/tex]
- [tex]\(5x^2 \cdot 6 = 30x^2\)[/tex]
- Multiply [tex]\(-x\)[/tex] by each term in [tex]\(2x + 6\)[/tex]:
- [tex]\(-x \cdot 2x = -2x^2\)[/tex]
- [tex]\(-x \cdot 6 = -6x\)[/tex]
- Multiply [tex]\(-3\)[/tex] by each term in [tex]\(2x + 6\)[/tex]:
- [tex]\(-3 \cdot 2x = -6x\)[/tex]
- [tex]\(-3 \cdot 6 = -18\)[/tex]
2. Combine all terms:
- List all the terms from the distribution:
[tex]\[
10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18
\][/tex]
3. Combine like terms:
- [tex]\(30x^2 - 2x^2 = 28x^2\)[/tex]
- [tex]\(-6x - 6x = -12x\)[/tex]
4. Write the final expression:
- Combine all simplified terms to get the final product:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
So the product of the polynomials [tex]\((5x^2 - x - 3)(2x + 6)\)[/tex] is:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
Therefore, the correct answer is option A: [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex].
1. Distribute each term of the first polynomial to the second polynomial:
- Multiply [tex]\(5x^2\)[/tex] by each term in [tex]\(2x + 6\)[/tex]:
- [tex]\(5x^2 \cdot 2x = 10x^3\)[/tex]
- [tex]\(5x^2 \cdot 6 = 30x^2\)[/tex]
- Multiply [tex]\(-x\)[/tex] by each term in [tex]\(2x + 6\)[/tex]:
- [tex]\(-x \cdot 2x = -2x^2\)[/tex]
- [tex]\(-x \cdot 6 = -6x\)[/tex]
- Multiply [tex]\(-3\)[/tex] by each term in [tex]\(2x + 6\)[/tex]:
- [tex]\(-3 \cdot 2x = -6x\)[/tex]
- [tex]\(-3 \cdot 6 = -18\)[/tex]
2. Combine all terms:
- List all the terms from the distribution:
[tex]\[
10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18
\][/tex]
3. Combine like terms:
- [tex]\(30x^2 - 2x^2 = 28x^2\)[/tex]
- [tex]\(-6x - 6x = -12x\)[/tex]
4. Write the final expression:
- Combine all simplified terms to get the final product:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
So the product of the polynomials [tex]\((5x^2 - x - 3)(2x + 6)\)[/tex] is:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
Therefore, the correct answer is option A: [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex].
