Factoring and Multiplying Polynomials

Study Notes

Multiplying Polynomials

Distribute each term in the first polynomial across all terms in the second polynomial.
Combine like terms after distribution to simplify the expression.

Factoring Polynomials

Given one factor, divide the polynomial by this factor to find the result.
Further factor any remaining polynomial completely.

Complete Factorization Steps

Identify two numbers that multiply to ( A \cdot C ) (product of leading coefficient and constant) and add to ( B ) (middle term).
Split the middle term using these two numbers to facilitate grouping.
Group terms into pairs and factor out common factors from each group.
If the remaining terms are identical, factor them out as well.

Difference of Squares

Applicable when there are only two terms that are perfect squares and a subtraction sign present (e.g. ( a^2 - b^2 )).
Recognize this as a specific case for simple factoring.

Factoring Difference of Squares

Take the square roots of both terms:
- The first term’s square root and the second term’s square root.
The expression factors as ( (\text{square root of } a + \text{square root of } b)(\text{square root of } a - \text{square root of } b) ).

Factoring Polynomials with Four Terms

Divide the polynomial into two groups of two terms each.
Factor out common factors from each group.
If the leftover terms are the same, factor them out and simplify further.
Check if any group can be factored using the difference of squares method.

Handling Multiple Polynomials

Maintain a separate factoring process for each polynomial in a sum or difference.
Factor each polynomial individually to achieve full factorization.

Polynomial Division

Write the polynomial equation ensuring each power of ( x ) is represented, inserting zeros where necessary.
Divide the leading term of the dividend by the leading term of the divisor.
Multiply the divisor by this term and subtract from the dividend.
Repeat the process until all terms are accounted for, ending with a possible remainder.

Finding Zeros of a Polynomial

Rearrange the polynomial to set it equal to zero.
Factor the polynomial and set each factor equal to zero to find solutions.
Solve each equation by adding or subtracting constants and dividing or multiplying coefficients as needed.

Factoring with Higher Powers

Treat ( x^4 ) as if it were ( x^2 ) for simplification.
Follow standard factoring procedures, translating the higher-degree polynomial into manageable components.

Description

This quiz covers essential techniques for multiplying and factoring polynomials, including distribution, complete factorization, and recognizing the difference of squares. Test your understanding of these fundamental concepts and improve your algebra skills.