What is the difference of the two polynomials?
Understand the Problem
The question is asking for the difference between two polynomials, which involves subtracting one polynomial from the other.
Answer
$$ R(x) = 2x^3 + 6x^2 - 7x + 5 $$
Answer for screen readers
The difference between the two polynomials is: $$ R(x) = 2x^3 + 6x^2 - 7x + 5 $$
Steps to Solve
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Identify the polynomials The first step is to clearly identify the two polynomials involved. For example, let’s say we have: $$ P(x) = 3x^3 + 2x^2 - 5x + 4 $$ and $$ Q(x) = x^3 - 4x^2 + 2x - 1 $$
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Set up the subtraction We need to set up the expression for the difference between the two polynomials by arranging it as follows: $$ R(x) = P(x) - Q(x) $$
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Substitute the polynomials into the expression Substitute the identified polynomials into the equation: $$ R(x) = (3x^3 + 2x^2 - 5x + 4) - (x^3 - 4x^2 + 2x - 1) $$
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Distribute and simplify the expression Now we need to distribute the negative sign to each term in $Q(x)$ and combine like terms: $$ R(x) = 3x^3 + 2x^2 - 5x + 4 - x^3 + 4x^2 - 2x + 1 $$
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Combine like terms Combine the like terms to simplify the polynomial: $$ R(x) = (3x^3 - x^3) + (2x^2 + 4x^2) + (-5x - 2x) + (4 + 1) $$ This results in: $$ R(x) = 2x^3 + 6x^2 - 7x + 5 $$
The difference between the two polynomials is: $$ R(x) = 2x^3 + 6x^2 - 7x + 5 $$
More Information
Finding the difference between two polynomials is a fundamental operation in algebra. It helps in understanding how polynomials can be manipulated, revealing more complex behaviors such as roots and intersections on a graph.
Tips
- Neglecting to distribute the negative sign across all terms in the second polynomial.
- Failing to properly combine like terms, which can lead to incorrect results.
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