Answer :
To find the product of the polynomials [tex]\((5x^2 - x - 3)(2x + 6)\)[/tex], we can expand the expression using the distributive property (often referred to as the FOIL method for binomials). Let's break it down:
1. Multiply each term of the first polynomial by each term of the second polynomial. This means we'll distribute each term in [tex]\((5x^2 - x - 3)\)[/tex] across [tex]\((2x + 6)\)[/tex].
2. Step-by-step distribution:
- First, multiply [tex]\(5x^2\)[/tex] by each term in [tex]\((2x + 6)\)[/tex]:
- [tex]\(5x^2 \times 2x = 10x^3\)[/tex]
- [tex]\(5x^2 \times 6 = 30x^2\)[/tex]
- Next, multiply [tex]\(-x\)[/tex] by each term in [tex]\((2x + 6)\)[/tex]:
- [tex]\(-x \times 2x = -2x^2\)[/tex]
- [tex]\(-x \times 6 = -6x\)[/tex]
- Finally, multiply [tex]\(-3\)[/tex] by each term in [tex]\((2x + 6)\)[/tex]:
- [tex]\(-3 \times 2x = -6x\)[/tex]
- [tex]\(-3 \times 6 = -18\)[/tex]
3. Combine all the terms obtained from the distribution:
- [tex]\(10x^3\)[/tex]
- [tex]\(30x^2 - 2x^2 = 28x^2\)[/tex] (Combine like terms)
- [tex]\(-6x - 6x = -12x\)[/tex] (Combine like terms)
- [tex]\(-18\)[/tex]
4. Write the final expression.
The expanded polynomial is:
[tex]\[ 10x^3 + 28x^2 - 12x - 18 \][/tex]
Thus, the correct answer is option B: [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex].
1. Multiply each term of the first polynomial by each term of the second polynomial. This means we'll distribute each term in [tex]\((5x^2 - x - 3)\)[/tex] across [tex]\((2x + 6)\)[/tex].
2. Step-by-step distribution:
- First, multiply [tex]\(5x^2\)[/tex] by each term in [tex]\((2x + 6)\)[/tex]:
- [tex]\(5x^2 \times 2x = 10x^3\)[/tex]
- [tex]\(5x^2 \times 6 = 30x^2\)[/tex]
- Next, multiply [tex]\(-x\)[/tex] by each term in [tex]\((2x + 6)\)[/tex]:
- [tex]\(-x \times 2x = -2x^2\)[/tex]
- [tex]\(-x \times 6 = -6x\)[/tex]
- Finally, multiply [tex]\(-3\)[/tex] by each term in [tex]\((2x + 6)\)[/tex]:
- [tex]\(-3 \times 2x = -6x\)[/tex]
- [tex]\(-3 \times 6 = -18\)[/tex]
3. Combine all the terms obtained from the distribution:
- [tex]\(10x^3\)[/tex]
- [tex]\(30x^2 - 2x^2 = 28x^2\)[/tex] (Combine like terms)
- [tex]\(-6x - 6x = -12x\)[/tex] (Combine like terms)
- [tex]\(-18\)[/tex]
4. Write the final expression.
The expanded polynomial is:
[tex]\[ 10x^3 + 28x^2 - 12x - 18 \][/tex]
Thus, the correct answer is option B: [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex].
