Problem 51 Subtract. $$ \begin{aligned}... [FREE SOLUTION]

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Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 5: Problem 51

Subtract. $$ \begin{aligned} &5 y^{3}-3 y^{2}\\\ &2 y^{3}+8 y^{2} \end{aligned} $$

Short Answer

Expert verified

The result is $ 3y^3 - 11y^2 $.

Step by step solution

Align the Polynomials Vertically

Align the polynomials so that the terms with the same degrees are in the same column: $ 5y^3 - 3y^2 $ $ 2y^3 + 8y^2 $

Subtract the Terms with the Same Degree

Subtract the corresponding terms of the polynomials: $ 5y^3 - 2y^3 = 3y^3 $ $ -3y^2 - 8y^2 = -11y^2 $

Combine the Results

Combine the results from Step 2 to form the resulting polynomial: The result is $ 3y^3 - 11y^2 $.

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Key Concepts

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

Polynomial Terms

When working with polynomials, understanding the terms is crucial. Each polynomial consists of terms, which are expressions separated by addition or subtraction. For example, in the polynomial $5y^3 - 3y^2$, there are two terms: $5y^3$ and $-3y^2$. Each term has a coefficient (the number in front) and a variable part (like $y^3$ or $y^2$). To effectively subtract polynomials, it's important to correctly identify and manage these terms. Look at the degree of each term, determined by the exponent on the variable. This helps in performing operations accurately.

Aligning Polynomials

Aligning polynomials means writing them so that terms with the same degree are at the same level. This makes it easier to see which terms will be subtracted. For the exercise above, we write:

$5y^3 - 3y^2$
$2y^3 + 8y^2$

Notice how $5y^3$ and $2y^3$ are lined up vertically because they have the same degree, 3. Similarly, $-3y^2$ and $8y^2$ are lined up because they have the same degree, 2. Proper alignment ensures accuracy in the subtraction process.

Combining Polynomials

After aligning the polynomials, the next step is subtraction. Subtract corresponding terms by degrees.

For the $y^3$ terms: $5y^3 - 2y^3 = 3y^3$.
For the $y^2$ terms: $-3y^2 - 8y^2 = -11y^2$.

The final step is to combine these subtracted terms into a single polynomial: $3y^3 - 11y^2$. This combination represents the result of the subtraction process. Each term in the final polynomial reflects the highest to the lowest degree, ensuring the polynomial is in standard form.

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91影视

Subtract. $$ \begin{aligned} &5 y^{3}-3 y^{2}\\\ &2 y^{3}+8 y^{2} \end{aligned} $$

Short Answer

Step by step solution

Align the Polynomials Vertically

Subtract the Terms with the Same Degree

Combine the Results

Key Concepts

Polynomial Terms

Aligning Polynomials

Combining Polynomials

One App. One Place for Learning.

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