Question

Use a vertical format or a horizontal format to add or subtract. $$\left(\frac{2}{5} a^{4}-2 a+7\right)-\left(-\frac{3}{10} a^{4}+6 a^{3}\right)-\left(2 a^{2}-7\right)$$

Short answer

The result of adding and subtracting the given polynomials is \( \frac{1}{2} a^{4} - 6a^{3} - 2a^{2} - 2a + 14 \)

Step-by-step solution

List down the given polynomials

The given polynomials are: \( \frac{2}{5} a^{4} - 2a + 7 \), \( - \frac{3}{10} a^{4} + 6a^{3} \), and \( 2a^{2} - 7 \).

Distributing the negative sign

Distribute the negative sign to each term in the polynomials that are being subtracted. This results in \( \frac{2}{5} a^{4} - 2a + 7 + \frac{3}{10} a^{4} - 6a^{3} - 2a^{2} + 7 \).

Adding like terms

Combine all similar terms together. This involves adding all \(a^{4}\) terms, \(a^{3}\) terms, \(a^{2}\) terms, \(a\) terms and constants together, respectively. This results in \( \frac{5}{10} a^{4} - 6a^{3} - 2a^{2} - 2a + 14 \).

Simplifying the coefficient of \(a^{4}\)

Simplify the fraction in the coefficient of \(a^{4}\). This leads to \( \frac{1}{2} a^{4} - 6a^{3} - 2a^{2} - 2a + 14 \).

Key concepts

Combining Like Terms

When working with polynomials, combining like terms is a critical step in simplification. Like terms have the same variable parts raised to the same power. For instance, \(2x^3\) and \(5x^3\) are like terms because they both contain the variable \(x\) raised to the third power.

To combine like terms, simply add or subtract the coefficients of those terms. Consider the terms \(2x^2\) and \(3x^2\); they can be combined to \(5x^2\) because they are like terms. On the other hand, \(4x^2\) and \(4x\) cannot be combined because they have different powers of \(x\).

In the given exercise involving polynomial subtraction, after distributing the negative signs, we identify and add together terms with the same power of \(a\). By carefully combining these like terms, the polynomial is simplified and more manageable for further operations.

Distributive Property

The distributive property allows you to multiply a single term and two or more terms inside a set of parentheses. This property is indispensable when dealing with polynomial subtraction, especially when subtracting entire polynomials from each other.

The standard form of the distributive property is \(a(b + c) = ab + ac\). When a negative sign precedes a set of parentheses, you must distribute it to every term inside. This effectively changes the sign of each term within the parentheses.

In our example, we apply the distributive property to subtract the polynomials. The negative signs in front of the parentheses are distributed across each term, transforming subtraction into addition of the opposite. For instance, \(-(-\frac{3}{10} a^4)\) becomes \(+\frac{3}{10} a^4\), as seen in Step 2 of the solution. This approach prevents errors and ensures that the subtraction is executed correctly across all the terms.

Simplifying Fractions

Simplifying fractions means reducing them to their lowest terms. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD). Simplified fractions are easier to work with, especially when combining coefficients in polynomial equations.

To illustrate, let’s take the fraction \(\frac{4}{8}\). The GCD of 4 and 8 is 4, so when we divide both by 4, we get \(\frac{1}{2}\), which is the simplified form of \(\frac{4}{8}\). Simpler fractions make it easier to identify like terms and conduct further operations.

In the exercise provided, Step 4 involves simplifying the fraction in the coefficient of \(a^4\). The operation transforms \(\frac{5}{10} a^4\) to its simplified form \(\frac{1}{2} a^4\). It's essential to always simplify fractions during polynomial operations to achieve the most simplified version of the expression.