Question

Use a horizontal format to add or subtract. $$\left(6 b^{4}-3 b^{3}-7 b^{2}+9 b+3\right)+\left(4 b^{4}-6 b^{2}+11 b-7\right)$$

Short answer

The addition of the two polynomial expressions is \(10b^4 - 3b^3 - 13b^2 + 20b - 4\).

Step-by-step solution

Aligning like terms

Arrange the expressions vertically so that like terms are aligned, i.e., the terms with the same powers of \(b\). This allows us to see clearly which terms are to be added or subtracted. \[ \begin{{align*}} &6 b^{4} -3 b^{3} -7 b^{2} +9 b +3 \ &4 b^{4} \quad \quad -6 b^{2} +11 b -7 \ \end{{align*}} \]

Addition/Subtraction of like terms

The next step is to add or subtract the coefficients of like terms. Be cautious about their signs. If a polynomial has no term of a certain degree, consider its coefficient as 0. The result is: \( (6b^4 + 4b^4) -3b^3 + (-7b^2 -6b^2) + (9b +11b) + (3-7) \)

Simplifying the Expression

After adding or subtracting the coefficients of the like terms, let's simplify the expression. The result will be: \( 10b^4 - 3b^3 - 13b^2 + 20b - 4 \)

Key concepts

Aligning Like Terms

When working with polynomial operations, one of the most essential techniques is the alignment of like terms. Like terms are terms that contain the same variable raised to the same power. For example, in the expression \(6b^4\) and \(4b^4\), both terms are like terms because they each contain the variable \(b\) raised to the fourth power.

To effectively add or subtract polynomials, it's important to organize these like terms in a vertical format. This helps to ensure that each term in one polynomial is matched with its corresponding term in the other polynomial. Imagine lining up numbers on an abacus where each row only accepts beads of the same color. When you arrange the polynomials in this manner, all the \(b^4\) terms are in one row, \(b^3\) in another, and so on. This visual presentation not only simplifies the process but also reduces the likelihood of computational errors.

Adding and Subtracting Polynomials

Polynomials are added or subtracted by combining their like terms. This process involves combining the coefficients of these like terms while keeping the variable and its exponent unchanged. The coefficient is the numerical part of a term, which lies before the variable.

For adding polynomials, if you have \(6b^4\) from one polynomial and \(4b^4\) from another, you add the coefficients (6 and 4) to get \(10b^4\). If you're subtracting, and you have \(6b^4\) and \(4b^4\), you subtract the second coefficient from the first (6 - 4) to get \(2b^4\). Remember, if there's no corresponding term in the other polynomial, you simply bring down the existing term. This process is similar to basic addition and subtraction, but now, you also need to pay attention to negative signs, as they can alter the result significantly.

Simplifying Polynomial Expressions

Simplifying polynomial expressions is the final step in the addition or subtraction process. After aligning and combining like terms, you'll often end up with a single polynomial expression that contains all of your results. Simplifying this expression means combining all the like terms correctly and presenting the polynomial in standard form, which typically goes from the highest power to the lowest.

This process could involve combining positive and negative coefficients and ensuring that the expression displays each term once. In our given exercise, we simplify to arrive at the expression \(10b^4 - 3b^3 - 13b^2 + 20b - 4\). It’s important to review this simplified expression to ensure that all like terms have been correctly combined and that no terms have been accidentally omitted.