Problem 46 - Elementary and Intermediate Algebra

Question

Use vertical form to add the polynomials. $$ \begin{array}{l} {3 x^{3}+4 x^{2}-3 x+5} \\ {3 x^{3}-4 x^{2}-x-7} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The resulting polynomial is \(6x^3 - 4x - 2\).

Step 1: Align the Terms

Align the polynomials so that like terms (terms with the same power of \(x\)) are in the same vertical column: \[\begin{array}{r}3x^3+4x^2-3x+5 \3x^3-4x^2-x-7 \\hline\end{array}\]

Step 2: Add the Cubic Terms

Add the cubic terms by summing the coefficients of \(x^3\). Here, the terms are \(3x^3\) and \(3x^3\). \[3x^3 + 3x^3 = 6x^3\]

Step 3: Add the Quadratic Terms

Add the quadratic terms by summing the coefficients of \(x^2\). The terms are \(4x^2\) and \(-4x^2\). \[4x^2 + (-4x^2) = 0\]

Step 4: Add the Linear Terms

Add the linear terms by summing the coefficients of \(x\). These are \(-3x\) and \(-x\). \[-3x + (-x) = -4x\]

Step 5: Add the Constant Terms

Add the constant terms by summing the constants 5 and -7. \[5 + (-7) = -2\]

Step 6: Write the Resulting Polynomial

Combine all the resulting terms from the previous steps to form the final polynomial. The result is:\[6x^3 + 0x^2 - 4x - 2\] Since the term with coefficient 0 can be omitted, the final answer is:\[6x^3 - 4x - 2\]

Key Concepts

Vertical Form

Polynomial addition using vertical form simplifies the process by aligning terms with the same degree, which are known as 'like terms'. In vertical form, you write the polynomials one under the other, ensuring that each column represents a different power of the variable, typically denoted as \(x\). This alignment makes it easier to add the coefficients of like terms as if you're adding numbers in standard arithmetic.

• Align all like terms in the same column.
• Add coefficients from top to bottom for each column.

By using vertical form, even complex expressions become easier to handle, streamlining the addition process. You simply focus on one column at a time, making no room for confusion about which terms to combine.

Cubic Terms

Cubic terms in a polynomial are those where the variable is raised to the third power, represented as \(x^3\). When adding polynomials, focus on the cubic terms first after aligning them vertically.

• Cubic terms have the form \(ax^3\), where \(a\) represents the coefficient.
• Sum the coefficients of all cubic terms. For instance, \(3x^3 + 3x^3 = 6x^3\).

Correctly adding cubic terms is crucial since they often define the leading term of the resulting polynomial, which influences its degree, a key characteristic of polynomials.

Quadratic Terms

Quadratic terms are those with the variable squared, such as \(x^2\). During addition, these terms must also be vertically aligned to correctly combine their coefficients.

• Quadratic terms look like \(bx^2\), with \(b\) as the coefficient.
• Even if the sum of their coefficients is zero, like \(4x^2\) and \(-4x^2\), still complete the addition to understand how they impact the polynomial.

While they may cancel each other out, handling them allows you to maintain a clear structure and ensures no mistakes are made in the summation process.

Linear Terms

Linear terms in a polynomial are those with the variable raised to the power of one, appearing in the form \(x\). When you add linear terms, align them as part of the vertical method and add their coefficients.

• Linear terms appear as \(cx\), where \(c\) is the coefficient.
• Follow the simple addition rule; for example, adding \(-3x\) and \(-x\) results in \(-4x\).

Summing linear terms gives balance to the polynomial. Even negative outcomes remain meaningfully integrated with the whole expression, easing transition into finding the final form.