Problem 51 Multiply the polynomials. $$(5... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

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Chapter 10: Problem 51

Multiply the polynomials. $$(5 y-1)\left(6 y^{2}+2 y+5\right)$$

Short Answer

Expert verified

30y^3 + 4y^2 + 23y - 5

Step by step solution

- Distribute the first term in the first polynomial

Multiply the first term of the first polynomial, which is 5y, by each term of the second polynomial (6y^2 + 2y + 5). This results in: 5y * 6y^2 + 5y * 2y + 5y * 5 which simplifies to: 30y^3 + 10y^2 + 25y.

- Distribute the second term in the first polynomial

Multiply the second term of the first polynomial, which is -1, by each term of the second polynomial (6y^2 + 2y + 5). This results in: -1 * 6y^2 + -1 * 2y + -1 * 5 which simplifies to: -6y^2 - 2y - 5.

- Combine like terms

Add the results from Step 1 and Step 2: 30y^3 + 10y^2 + 25y - 6y^2 - 2y - 5. Combine the like terms: (30y^3) + (10y^2 - 6y^2) + (25y - 2y) - 5, which simplifies to: 30y^3 + 4y^2 + 23y - 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property

To multiply polynomials, the distributive property is a crucial technique. This property allows us to multiply each term of one polynomial by every term of another. This can be written as:

$a(b + c) = ab + ac$.

In our exercise, we multiply the polynomials $(5y-1)(6y^2 + 2y + 5)$.

We start by distributing each term of $5y-1$ to every term in $6y^2 + 2y + 5$. First, we distribute $5y$:

$5y * 6y^2 = 30y^3$
$5y * 2y = 10y^2$
$5y * 5 = 25y$

Next, we distribute $-1$:

$-1 * 6y^2 = -6y^2$
$-1 * 2y = -2y$
$-1 * 5 = -5$

This technique ensures no term is missed, and helps avoid errors.

Combining Like Terms

After using the distributive property, we often have many terms to handle. Some of these terms are similar and can be combined. Like terms have the same variables raised to the same power. For example, $10y^2$ and $-6y^2$ are like terms.

In our example, after distributing, we get:
$30y^3 + 10y^2 + 25y - 6y^2 - 2y - 5$.

Now, we identify and combine like terms:

The $y^2$ terms: $10y^2 - 6y^2 = 4y^2$
The $y$ terms: $25y - 2y = 23y$
Constant terms remain as is

This results in: $30y^3 + 4y^2 + 23y - 5$.

Combining like terms simplifies the polynomial to its most reduced form, making it easier to understand and work with.

Polynomial Simplification

Simplification is the final step in polynomial multiplication. It involves combining like terms and ensuring the polynomial is in its simplest form. This process makes subsequent calculations easier.

In the given problem, after distributing and combining terms, we achieve a simplified polynomial:
$30y^3 + 4y^2 + 23y - 5$. Simplification helps in:

Identifying and grouping like terms
Reducing the polynomial to a more manageable form

By the end of simplification, the polynomial is neat and ready for further operations. Understanding these steps is key to mastering polynomial multiplication and various algebraic processes.

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91影视

Multiply the polynomials. $$(5 y-1)\left(6 y^{2}+2 y+5\right)$$

Short Answer

Step by step solution

- Distribute the first term in the first polynomial

- Distribute the second term in the first polynomial

- Combine like terms

Key Concepts

Distributive Property

Combining Like Terms

Polynomial Simplification

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