Multiplication of polynomials

Definition

Multiplication of polynomials involves combining two or more polynomial expressions to create a single polynomial. This process often requires using the distributive property, where each term in one polynomial is multiplied by each term in the other polynomial, leading to a sum of products. Mastering this skill is essential for simplifying expressions and solving equations that involve polynomials.

5 Must Know Facts For Your Next Test

1. When multiplying polynomials, it's important to remember to apply the distributive property correctly to ensure all terms are accounted for.
2. The product of two polynomials can lead to an increase in the degree of the resulting polynomial, specifically the degree will be the sum of the degrees of the original polynomials.
3. Combining like terms after multiplying is crucial for simplifying the resulting polynomial expression.
4. The FOIL method is a special case used for multiplying two binomials, representing First, Outside, Inside, and Last products.
5. Multiplying polynomials is foundational for factoring, as it helps in reversing the multiplication process to express a polynomial in its factored form.

Review Questions

• Explain how the distributive property is applied when multiplying two binomials together.
  • When multiplying two binomials, you apply the distributive property by multiplying each term of the first binomial by each term of the second binomial. For example, if you have (a + b)(c + d), you multiply 'a' by both 'c' and 'd', then 'b' by both 'c' and 'd'. This results in four products: ac, ad, bc, and bd, which you then combine to form the final expression: ac + ad + bc + bd.
• What is the significance of combining like terms after multiplying polynomials, and how does it affect the final expression?
  • Combining like terms after multiplying polynomials is crucial because it simplifies the final expression into its simplest form. Each term with the same variable and exponent can be grouped together to reduce clutter and improve clarity. For example, after multiplying (x + 2)(x + 3), you would get x^2 + 5x + 6. By combining like terms where applicable, you ensure that your polynomial accurately reflects all contributions from the multiplication process.
• Evaluate how understanding the multiplication of polynomials assists in solving quadratic equations and factoring them back into their constituent parts.
  • Understanding multiplication of polynomials is key when solving quadratic equations since quadratics can often be expressed as products of binomials. For instance, if you factor x^2 + 5x + 6 into (x + 2)(x + 3), knowing how to multiply these binomials confirms that they equate back to the original quadratic. This skill not only helps solve equations but also allows for easier graphing and understanding of polynomial behavior by identifying roots through factored forms.

Related terms

Binomial:

A polynomial that contains exactly two terms, which can be combined through addition or subtraction.

Trinomial: A polynomial with three distinct terms, commonly used in expressions and equations.

Degree of a Polynomial: The highest power of the variable in a polynomial, which determines its classification and behavior.