Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 18: Polynomials

Lesson 1: Addition and Subtraction

In this Grade 4 AMC Math lesson from AoPS: Introduction to Algebra, students learn how to add and subtract polynomials by combining like terms, including expressions with degrees up to cubic and beyond. The lesson introduces subscript notation for polynomial coefficients and the general form of a degree-n polynomial. Students also practice determining unknown constants that change a polynomial's degree by canceling leading terms.

Section 1

General Polynomial Form with Subscript Notation

Property

A polynomial expression has the form

anxn+an−1xn−1+an−2xn−2+⋯+a2x2+a1x+a0a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots + a_2 x^2 + a_1 x + a_0

where a0,a1,a2,…,ana_0, a_1, a_2, \ldots, a_n are constants and an≠0a_n \neq 0. The coefficient ana_n of the highest power term is called the lead coefficient. Polynomials can be written in descending powers, where terms are ordered from the highest degree to the lowest, or in ascending powers, where terms are ordered from lowest degree to highest.

Examples

Section 2

Polynomials

Property

• A polynomial is a sum of terms, each of which is a power of a variable with a constant coefficient and a whole number exponent.

• The degree of a polynomial in one variable is the largest exponent that appears in any term.

• Like terms are any terms that are exactly alike in their variable factors. The exponents on the variable factors must also match.

Section 3

Add and Subtract Polynomials

Property

To add or subtract polynomials, combine like terms. Like terms are monomials that have the same variables with the same exponents. The Commutative Property allows rearranging terms to group like terms together. When subtracting polynomials, be careful to distribute the negative sign to every term in the polynomial being subtracted.

Examples

  • To find the sum: (3x2+5x−2)+(x2−2x+7)=(3x2+x2)+(5x−2x)+(−2+7)=4x2+3x+5(3x^2 + 5x - 2) + (x^2 - 2x + 7) = (3x^2 + x^2) + (5x - 2x) + (-2 + 7) = 4x^2 + 3x + 5.
  • To find the difference: (8a2−4a+1)−(3a2−a−5)=8a2−4a+1−3a2+a+5=5a2−3a+6(8a^2 - 4a + 1) - (3a^2 - a - 5) = 8a^2 - 4a + 1 - 3a^2 + a + 5 = 5a^2 - 3a + 6.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

General Polynomial Form with Subscript Notation

Property

A polynomial expression has the form

anxn+an−1xn−1+an−2xn−2+⋯+a2x2+a1x+a0a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots + a_2 x^2 + a_1 x + a_0

where a0,a1,a2,…,ana_0, a_1, a_2, \ldots, a_n are constants and an≠0a_n \neq 0. The coefficient ana_n of the highest power term is called the lead coefficient. Polynomials can be written in descending powers, where terms are ordered from the highest degree to the lowest, or in ascending powers, where terms are ordered from lowest degree to highest.

Examples

Section 2

Polynomials

Property

• A polynomial is a sum of terms, each of which is a power of a variable with a constant coefficient and a whole number exponent.

• The degree of a polynomial in one variable is the largest exponent that appears in any term.

• Like terms are any terms that are exactly alike in their variable factors. The exponents on the variable factors must also match.

Section 3

Add and Subtract Polynomials

Property

To add or subtract polynomials, combine like terms. Like terms are monomials that have the same variables with the same exponents. The Commutative Property allows rearranging terms to group like terms together. When subtracting polynomials, be careful to distribute the negative sign to every term in the polynomial being subtracted.

Examples

  • To find the sum: (3x2+5x−2)+(x2−2x+7)=(3x2+x2)+(5x−2x)+(−2+7)=4x2+3x+5(3x^2 + 5x - 2) + (x^2 - 2x + 7) = (3x^2 + x^2) + (5x - 2x) + (-2 + 7) = 4x^2 + 3x + 5.
  • To find the difference: (8a2−4a+1)−(3a2−a−5)=8a2−4a+1−3a2+a+5=5a2−3a+6(8a^2 - 4a + 1) - (3a^2 - a - 5) = 8a^2 - 4a + 1 - 3a^2 + a + 5 = 5a^2 - 3a + 6.