The purpose of this lesson is to introduce students to the idea of closure (although not necessarily the word “closure,” unless students ask if there is a word for this idea), and to show that both integers and polynomials are closed under addition, subtraction, and multiplication. Students will also practice adding, subtracting, and multiplying polynomials, with the goal of finding out whether the result will always be a polynomial. Based on their findings, they willl make arguments for what they think will happen, and critique each other’s arguments (MP3). Students will deepen their understanding of what polynomials are and what they are not, and of the similarities between integers and polynomials.
The work of this lesson connects to upcoming work because students are left with the question of what happens when one polynomial is divided by another, and they will see in future lessons what the possibilities are. A robust understanding of how to multiply polynomials is key for successful division, and starting with this lesson, students have opportunities to practice multiplying polynomials in order to identify strategies that work best for them.
- Comprehend that when polynomials are combined by addition, subtraction, or multiplication, the result is a polynomial.
- Justify (orally) conclusions about what happens when integers or polynomials are combined using arithmetic operations.
- Let's do arithmetic with polynomials.
- I understand that if you add, subtract, or multiply polynomials, you get another polynomial.
The degree of a polynomial in \(x\) is the highest exponent occuring on \(x\) when you write the polynomial out as a sum of non-zero constants times powers of \(x\) (with like terms collected).
A polynomial function of \(x\) is a function given by a sum of terms, each of which is a constant times a whole number power of \(x\). The word polynomial is used to refer both to the function and to the expression defining it.
A point on the graph of a function that is higher than any of the points around it.
A point on the graph of a function that is lower than any of the points around it.