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Chapter 6: Problem 457

Divide each polynomial by the monomial. $$\frac{51 y^{4}+42 y^{2}}{3 y^{2}}$$

Short Answer

Expert verified
17y^2 + 14

Step by step solution

01

Separate the terms

Divide each term of the polynomial in the numerator by the monomial in the denominator. This will give: \( \frac{51 y^{4}}{3 y^{2}} + \frac{42 y^{2}}{3 y^{2}} \).
02

Simplify each fraction

Simplify each fraction separately: The first term: \( \frac{51 y^{4}}{3 y^{2}} = 17 y^{4-2} = 17 y^{2} \). The second term: \( \frac{42 y^{2}}{3 y^{2}} = 14 y^{2-2} = 14 \).
03

Combine the simplified terms

Combine the simplified terms to get the final answer: \( 17 y^{2} + 14 \).

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Key Concepts

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

monomial division
When dividing polynomials by a monomial, we break down the process into smaller, manageable parts.

First, separate the terms in the polynomial by the monomial in the denominator.

For instance, consider the equation from the exercise: \(\frac{51 y^{4} + 42 y^{2}}{3 y^{2}} \).

This can be broken down into: \(\frac{51 y^{4}}{3 y^{2}} + \frac{42 y^{2}}{3 y^{2}} \).

Now, each term in the numerator is divided separately by the denominator.

Remember, division of terms can be simplified by dividing the coefficients (numbers) and subtracting the exponents of like variables.
simplifying fractions
Simplifying fractions is an essential skill in algebra.

In the context of polynomial division, simplify each term by dividing the top (numerator) by the bottom (denominator).

Using our exercise:
  • The first term is \(\frac{51 y^{4}}{3 y^{2}} \). Simplifying, this gives \(\frac{51}{3} y^{4-2} = 17 y^{2} \).
  • The second term is \(\frac{42 y^{2}}{3 y^{2}} \). Simplifying, this gives \(\frac{42}{3} y^{2-2} = 14 \).
Note that subtracting the exponents follows the rule \( \frac{a^m}{a^n} = a^{m-n} \).
Always double-check arithmetic to make sure your fractions are simplified fully.
polynomial operations
Polynomial operations, such as addition, subtraction, and division, are foundational concepts in algebra.

When handling polynomial division, focus on performing systematic steps:
  • First, separate terms to make the polynomial more manageable.
  • Second, simplify fractions resulting from this separation.
  • Finally, recombine the simplified terms to get your final answer.
In the problem provided, we divided, simplified, and recombined terms:

After separating and simplifying, the operation \(\frac{51 y^{4} + 42 y^{2}}{3 y^{2}}\text{ gives } 17 y^{2} + 14.\)

This method ensures you effectively handle polynomial divisions without missing any critical steps.

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Most popular questions from this chapter

Simplify. (a) \(\frac{1}{c^{-5}}\) (b) \(\frac{1}{5^{-2}}\)

Divide each polynomial by the binomial. $$\left(a^{2}-2 a-35\right) \div(a+5)$$

Simplify. (a) \(3^{-4}\) (b) \(10^{-2}\)

Divide the monomials. $$\frac{45 x^{5} y^{9}}{-60 x^{8} y^{6}}$$

Divide each polynomial by the binomial. $$\left(x^{3}-216\right) \div(x-6)$$
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