Question

Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. $$ \left(x^{3}-2 x^{2}+5 x+10\right)-\left(2 x^{2}-4 x+3\right) $$

Short answer

The polynomial in standard form is \( x^3 - 4x^2 + 9x + 7 \).

Step-by-step solution

Distribute the negative sign

Distribute the negative sign to each term inside the second parenthesis: \[ (x^{3} - 2x^{2} + 5x + 10) - (2x^{2} - 4x + 3) \] becomes \[ x^{3} - 2x^{2} + 5x + 10 - 2x^{2} + 4x - 3 \]

Combine like terms

Group and combine the like terms: \( -2x^{2} \) and \( -2x^{2} \) combine to \( -4x^{2} \), \( 5x \) and \( 4x \) combine to \( 9x \), and \( 10 \) and \( -3 \) combine to \( 7 \). So, \[ x^{3} - 2x^{2} + 5x + 10 - 2x^{2} + 4x - 3 \] becomes \[ x^{3} - 4x^{2} + 9x + 7 \].

Write final answer in standard form

Ensure that the polynomial is written in standard form, which arranges terms in descending order of exponents: \[ x^{3} - 4x^{2} + 9x + 7 \].

Key concepts

Adding Polynomials

To add polynomials, simply combine like terms from each polynomial. Like terms are terms that have the same variable raised to the same power. For example, if we had the polynomials \(3x^2 + 4x + 5\) and \(2x^2 - x + 7\), we would add the coefficients of the like terms together: \(3x^2\) and \(2x^2\) combine to form \(5x^2\), \(4x\) and \(-x\) combine to form \(3x\), and the constants \(5\) and \(7\) combine to form \(12\). Thus, the result is \(5x^2 + 3x + 12\).

Subtracting Polynomials

Subtracting polynomials follows a similar method to adding them, but with an additional step. You need to distribute the negative sign across each term in the polynomial being subtracted. For example, to subtract \((2x^2 - 4x + 3)\) from \((x^3 - 2x^2 + 5x + 10)\), you first distribute the negative:

• The polynomial \((2x^2 - 4x + 3)\) becomes \(-2x^2 + 4x - 3\).
• Then, combine like terms: \(x^3\) remains \(x^3\), \(-2x^2\) and \(-2x^2\) becomes \(-4x^2\), \(5x\) and \(4x\) becomes \(9x\), and \(10\) and \(-3\) becomes \(7\).

In this manner, the expression is simplified to \(x^3 - 4x^2 + 9x + 7\).

Combining Like Terms

Combining like terms means to sum the coefficients of terms with the same variable and exponent. This is crucial in simplifying polynomials. For example, in the expression \(x^3 - 2x^2 + 5x + 10 - 2x^2 + 4x - 3\), look for terms with the same exponents:

• The \(-2x^2\) terms combine to give \(-4x^2\).
• The \(5x\) and \(4x\) terms combine to give \(9x\).
• The constants \(10\) and \(-3\) combine to give \(7\).

Combining like terms simplifies the polynomial to \(x^3 - 4x^2 + 9x + 7\).

Polynomial Standard Form

Writing a polynomial in standard form means arranging the terms in descending order of their exponents. This makes your work more organized and easy to understand. For example, if we have a polynomial \(x^{3} - 4x^{2} + 9x + 7\), it is already in standard form because the terms are positioned from the highest exponent (\(x^3\)) to the lowest (constant term: \(7\)). Always ensure your final answer is in this form for clarity and correctness.