College Algebra Study Guide: Polynomials – Standard Form, Classification, Addition & Subtraction

Q1. Write each polynomial in standard form: $x^2 + x + 3$

Background

Topic: Polynomials – Standard Form

This question tests your ability to arrange the terms of a polynomial in standard form, which means writing the terms in descending order of degree.

Key Terms:

• Standard Form: A polynomial written with terms in order from highest to lowest degree.

• Degree: The highest exponent of the variable in the polynomial.

Step-by-Step Guidance

1. Identify the degree of each term: $x^2$ (degree 2), $x$ (degree 1), $3$ (degree 0).

2. Arrange the terms in descending order of degree: Start with the highest degree term.

3. Write the polynomial in standard form by listing the terms from highest to lowest degree.

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Q2. Write each polynomial in standard form: $7k^2 + 64k^3 - 2k$

Background

Topic: Polynomials – Standard Form

This question asks you to rewrite the polynomial so that the terms are ordered by descending degree.

Key Terms:

• Standard Form: Terms are written from highest to lowest degree.

• Degree: The exponent of the variable in each term.

Step-by-Step Guidance

1. Identify the degree of each term: $64k^3$ (degree 3), $7k^2$ (degree 2), $-2k$ (degree 1).

2. Order the terms from highest to lowest degree.

3. Write the polynomial in standard form.

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Q3. Classify the polynomial $7p^2 + 3p$ by degree and number of terms.

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Topic: Classifying Polynomials

This question tests your understanding of how to classify polynomials based on their degree and the number of terms.

Key Terms:

• Degree: The highest exponent in the polynomial.

• Number of Terms: Count each separate term in the polynomial.

Step-by-Step Guidance

1. Identify the degree of each term: $7p^2$ (degree 2), $3p$ (degree 1).

2. The degree of the polynomial is the highest degree present.

3. Count the number of terms in the polynomial.

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Q4. Find the sum: $(x^2 - 4x + 3) + (3x^2 - 3x - 5)$

Background

Topic: Adding Polynomials

This question tests your ability to add two polynomials by combining like terms.

Key Terms and Formulas:

• Like Terms: Terms with the same variable and exponent.

• Addition of Polynomials: Combine coefficients of like terms.

Step-by-Step Guidance

1. Write both polynomials in standard form: $x^2 - 4x + 3$ and $3x^2 - 3x - 5$.

2. Group like terms together: $x^2$ terms, $x$ terms, and constants.

3. Add the coefficients of each group: $x^2$ terms, $x$ terms, and constants.

4. Write the resulting polynomial in standard form.

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Q5. Subtract: $(8r^2 - 12r + 4) - (3r^2 + 5r - 1)$

Background

Topic: Subtracting Polynomials

This question tests your ability to subtract one polynomial from another by combining like terms and changing the signs appropriately.

Key Terms and Formulas:

• Subtraction of Polynomials: Distribute the negative sign and combine like terms.

• Like Terms: Terms with the same variable and exponent.

Step-by-Step Guidance

1. Write both polynomials in standard form: $8r^2 - 12r + 4$ and $3r^2 + 5r - 1$.

2. Distribute the negative sign to each term in the second polynomial.

3. Group like terms together: $r^2$ terms, $r$ terms, and constants.

4. Subtract the coefficients of each group.

Try solving on your own before revealing the answer!