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Find the product.

[tex](y+6)\left(2 y^3+8 y^2-18 y+2\right)[/tex]

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GinnyAnsweridn
To find the product [tex]\((y+6)\left(2 y^3 + 8 y^2 - 18 y + 2\right)\)[/tex], we will expand the expression step-by-step.

1. Distribute each term in [tex]\(y + 6\)[/tex] through the polynomial [tex]\(2 y^3 + 8 y^2 - 18 y + 2\)[/tex].

First, distribute [tex]\(y\)[/tex] through each term of the polynomial:

[tex]\[ y \cdot (2 y^3 + 8 y^2 - 18 y + 2) \][/tex]

This yields:

[tex]\[ y \cdot 2 y^3 = 2 y^4 \][/tex]
[tex]\[ y \cdot 8 y^2 = 8 y^3 \][/tex]
[tex]\[ y \cdot (-18 y) = -18 y^2 \][/tex]
[tex]\[ y \cdot 2 = 2 y \][/tex]

So, combining these results, we get:

[tex]\[ y \cdot (2 y^3 + 8 y^2 - 18 y + 2) = 2 y^4 + 8 y^3 - 18 y^2 + 2 y \][/tex]

Next, distribute [tex]\(6\)[/tex] through each term of the polynomial:

[tex]\[ 6 \cdot (2 y^3 + 8 y^2 - 18 y + 2) \][/tex]

This yields:

[tex]\[ 6 \cdot 2 y^3 = 12 y^3 \][/tex]
[tex]\[ 6 \cdot 8 y^2 = 48 y^2 \][/tex]
[tex]\[ 6 \cdot (-18 y) = -108 y \][/tex]
[tex]\[ 6 \cdot 2 = 12 \][/tex]

So, combining these results, we get:

[tex]\[ 6 \cdot (2 y^3 + 8 y^2 - 18 y + 2) = 12 y^3 + 48 y^2 - 108 y + 12 \][/tex]

2. Combine all the resulting terms from both distributions.

Adding together the terms obtained from distributing [tex]\(y\)[/tex] and [tex]\(6\)[/tex]:

[tex]\[ 2 y^4 + 8 y^3 - 18 y^2 + 2 y \][/tex]
[tex]\[ + 12 y^3 + 48 y^2 - 108 y + 12 \][/tex]

Combine like terms:

- [tex]\(2 y^4\)[/tex] (no like terms, remains as is)
- [tex]\(8 y^3 + 12 y^3 = 20 y^3\)[/tex]
- [tex]\(-18 y^2 + 48 y^2 = 30 y^2\)[/tex]
- [tex]\(2 y - 108 y = -106 y\)[/tex]
- [tex]\(12\)[/tex] (no like terms, remains as is)

Thus, the final product is:

[tex]\[ 2 y^4 + 20 y^3 + 30 y^2 - 106 y + 12 \][/tex]

So, the expanded form of the product [tex]\((y+6)\left(2 y^3 + 8 y^2 - 18 y + 2\right)\)[/tex] is:

[tex]\[ 2 y^4 + 20 y^3 + 30 y^2 - 106 y + 12 \][/tex]

Answer: 2y⁴ + 20y³- 66y² -106y + 12

Step-by-step explanation:

By using distribution, you can find the product of this equation. First, let's multiply y by the polynomial:

(y) (2y³ +8y² - 18y +2):

  • y × 2y³ = 2y⁴
  • y × 8y²= 8y³
  • y × -18y = -18y²
  • y × 2 = 2y

The polynomial multiplied by y = 2y⁴ + 8y³ - 18y² +2y

Now, multiply 6 by the polynomial: (6) (2y³ +8y² - 18y +2):

  • 6 × 2y³ = 12y³
  • 6 × 8y²= 48y²
  • 6 × -18y = -108y
  • 6 × 2 = 12

The polynomial multiplied by 6= 12y³ + 48y² - 108y + 12

Combine your two final equations to get:

(2y⁴ + 8y³ - 18y² +2y) + (12y³ + 48y² - 108y + 12)

Combine like terms to get your final answer, which is

2y⁴ + 20y³- 30y² -106y + 12

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