Addition, Subtraction, and Multiplication of Polynomials Section R.3 Addition, Subtraction, and Multiplication of Polynomials

R.3 Addition, Subtraction, and Multiplication of Polynomials Identify the terms, coefficients, and degree of a polynomial. Add, subtract, and multiply polynomials.

Polynomials Polynomials are a type of algebraic expression. Some examples of polynomials are

Polynomials in One Variable A polynomial in one variable is any expression of the type where n is a nonnegative integer, an,…, a0 are real numbers, called coefficients. The parts of the polynomial separated by plus signs are called terms. The leading coefficient is an, and the constant term is a0. If an  0, the degree of the polynomial is n. The polynomial is said to be written in descending order, because the exponents decrease from left to right.

Example Identify the terms of the polynomial. 2x4  7.5x3 + x  12 The terms are: 2x4, 7.5x3, x, and 12.

Degree of a Polynomial Algebraic expressions like 3ab3 – 8 are a polynomial in several variables. The degree of a term is the sum of the exponents of the variables in that term. The degree of a polynomial is the degree of the term of the highest degree. The degrees of the terms of 3ab3 – 8 are 3 and 0. The degree of the polynomial is 3.

Addition and Subtraction If two terms of an expression have the same variables raised to the same powers, they are called like terms, or similar terms. We can combine, or collect, like terms using the distributive property. For example, 3y2 and 5y2 are like terms and We add or subtract polynomials by combining like terms.

Example Add: Subtract: We can subtract by adding an opposite:

Multiplication Multiplication is based on the distributive property. Example Multiply:

Multiplication In general, to multiply two polynomials, we multiply each term of one by each term of the other and add the products. Example Multiply:

Multiplication - FOIL We can find the product of two binomials by multiplying the First terms, then the Outer terms, then the Inner terms, then the Last terms. Then we combine like terms if possible. This procedure is called FOIL. Example: Multiply (2x  5)(3x +4) (x  5)(4x  1) = 4x2 – x – 20x + 5 = 6x2 – 13x – 28 F O I L

Special Products of Binomials (A + B)2 = A2 + 2AB + B2 Square of a sum (A – B)2 = A2 – 2AB + B2 Square of a difference (A + B)(A – B) = A2 – B2 Product of a sum and a difference

Example a) (4x +1)2 = (4x)2 + 2 • 4x + 12 = 16x2 +8x + 1 b) c)