Problem 20 - Algebra for College Students

Question

Add the given polynomials. \(-x^{2}-x-4,2 x^{2}-7 x+9\), and \(-3 x^{2}+6 x-10\)

Step-by-Step Solution

Answer

The sum of the polynomials is \(-2x^{2} - 2x - 5\).

Step 1: Arrange the Polynomials

Write down the polynomials with like terms aligned: \(-x^{2}\ - x\ - 4\)\(2x^{2}\ - 7x\ + 9\)\(-3x^{2}\ + 6x\ - 10\)

Step 2: Combine Like Terms for the Quadratic Terms

Focus on the quadratic terms: \(-x^{2}\), \(2x^{2}\), and \(-3x^{2}\).Combine these terms: \(-1 + 2 - 3 = -2\).The resulting quadratic term is \(-2x^{2}\).

Step 3: Combine Like Terms for the Linear Terms

Focus on the linear terms: \(-x\), \(-7x\), and \(6x\).Combine these terms: \(-1 - 7 + 6 = -2\).The resulting linear term is \(-2x\).

Step 4: Combine Like Terms for the Constant Terms

Focus on the constants: \(-4\), \(9\), and \(-10\).Combine these terms: \(-4 + 9 - 10 = -5\).The resulting constant is \(-5\).

Step 5: Write the Final Expression

Combine the results from the previous steps to write the final expression:\(-2x^{2} - 2x - 5\).

Key Concepts

Quadratic Terms

Quadratic terms refer to the parts of a polynomial which involve the square of the variable. In a general form, quadratic terms look like this: \(ax^2\), where \(a\) is a coefficient, and \(x^2\) indicates that the variable \(x\) is squared. These terms are crucial when dealing with quadratic equations, which typically follow the form \(ax^2 + bx + c\).
When adding polynomials, it's important to align and combine these quadratic terms. For example, if you have multiple polynomials, you simply sum up the coefficients of the \(x^2\) terms:

• In the given exercise, the quadratic terms are \(-x^2\), \(2x^2\), and \(-3x^2\).
• Add these coefficients: \(-1 + 2 - 3 = -2\).

This operation results in a new quadratic term \(-2x^2\). Understanding the sign of each term is vital to carrying out the addition correctly.

Linear Terms

Linear terms in a polynomial involve the variable raised to the power of one, appearing as \(bx\), where \(b\) is the coefficient. These terms form part of linear equations, typically seen as \(ax + b\). Linear terms reflect the nature of a polynomial's first-degree impact.

• For the exercise, linear terms identified were \(-x\), \(-7x\), and \(6x\).
• By combining their coefficients, we apply: \(-1 - 7 + 6 = -2\).

The result is \(-2x\), which indicates the collective effect of the variable \(x\) in all the given polynomials. These terms significantly influence how a polynomial behaves, especially its slope when graphed alongside quadratic and constant terms.

Constant Terms

Constant terms are the static, number-only components of polynomials, represented by \(c\) in the format \(ax^2 + bx + c\). These numbers do not involve any variable and remain unchanged when the variable changes, providing the value of the expression when the variable equals zero.
In the context of polynomial addition, constant terms are straightforward to combine.

• For instance, the constants in the exercise are \(-4\), \(9\), and \(-10\).
• Adding these: \(-4 + 9 - 10 = -5\).

Thus, the result for the constant terms is \(-5\). Constant terms might seem simple, but they form the base height of the polynomial graph along the vertical axis, influencing where the graph starts.