Answer :
To find the product of the polynomials [tex]\((5x^2 - x - 3)(2x + 6)\)[/tex], we can use the distributive property (also known as the FOIL method for binomials). Here's how you break it down step-by-step:
1. Distribute each term in the first polynomial to each term in the second polynomial:
- Multiply [tex]\(5x^2\)[/tex] by each term in the second polynomial:
- [tex]\(5x^2 \cdot 2x = 10x^3\)[/tex]
- [tex]\(5x^2 \cdot 6 = 30x^2\)[/tex]
- Multiply [tex]\(-x\)[/tex] by each term in the second polynomial:
- [tex]\(-x \cdot 2x = -2x^2\)[/tex]
- [tex]\(-x \cdot 6 = -6x\)[/tex]
- Multiply [tex]\(-3\)[/tex] by each term in the second polynomial:
- [tex]\(-3 \cdot 2x = -6x\)[/tex]
- [tex]\(-3 \cdot 6 = -18\)[/tex]
2. Combine all the terms:
- [tex]\(10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18\)[/tex]
3. Combine like terms:
- The terms [tex]\(30x^2\)[/tex] and [tex]\(-2x^2\)[/tex] combine to [tex]\(28x^2\)[/tex].
- The terms [tex]\(-6x\)[/tex] and [tex]\(-6x\)[/tex] combine to [tex]\(-12x\)[/tex].
So the final expanded form of the product is:
[tex]\[10x^3 + 28x^2 - 12x - 18\][/tex]
Therefore, the correct answer is B: [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex].
1. Distribute each term in the first polynomial to each term in the second polynomial:
- Multiply [tex]\(5x^2\)[/tex] by each term in the second polynomial:
- [tex]\(5x^2 \cdot 2x = 10x^3\)[/tex]
- [tex]\(5x^2 \cdot 6 = 30x^2\)[/tex]
- Multiply [tex]\(-x\)[/tex] by each term in the second polynomial:
- [tex]\(-x \cdot 2x = -2x^2\)[/tex]
- [tex]\(-x \cdot 6 = -6x\)[/tex]
- Multiply [tex]\(-3\)[/tex] by each term in the second polynomial:
- [tex]\(-3 \cdot 2x = -6x\)[/tex]
- [tex]\(-3 \cdot 6 = -18\)[/tex]
2. Combine all the terms:
- [tex]\(10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18\)[/tex]
3. Combine like terms:
- The terms [tex]\(30x^2\)[/tex] and [tex]\(-2x^2\)[/tex] combine to [tex]\(28x^2\)[/tex].
- The terms [tex]\(-6x\)[/tex] and [tex]\(-6x\)[/tex] combine to [tex]\(-12x\)[/tex].
So the final expanded form of the product is:
[tex]\[10x^3 + 28x^2 - 12x - 18\][/tex]
Therefore, the correct answer is B: [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex].
