University:
California State University, Northridge
Course:
MATH 320 | Foundations of Higher Mathematics
Academic year:

2024
Views:

392

Pages:

11
Author:

boonepanicrtn

§ 12.3 Introduction to Polynomials Polynomial Vocabulary Term – a number or a product of a number and variables raised to powers Coefficient – numerical factor of a term Constant – term which is only a number Polynomial is a sum of terms involving variables raised to a whole number exponent, with no variables appearing in any denominator. 29 Polynomial Vocabulary In the polynomial 7x5 + x2y2 – 4xy + 7 There are 4 terms: 7x5, x2y2, -4xy and 7. The coefficient of term 7x5 is 7, of term x2y2 is 1, of term –4xy is –4 and of term 7 is 7. 7 is a constant term. 30 Types of Polynomials Monomial is a polynomial with 1 term. Binomial is a polynomial with 2 terms. Trinomial is a polynomial with 3 terms. 31 Degrees Degree of a term To find the degree, take the sum of the exponents on the variables contained in the term. Degree of a constant is 0. Degree of the term 5a4b3c is 8 (remember that c can be written as c1). Degree of a polynomial To find the degree, take the largest degree of any term of the polynomial. Degree of 9x3 – 4x2 + 7 is 3. 32 Evaluating Polynomials Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved. Example Find the value of 2x3 – 3x + 4 when x = −2. 2x3 – 3x + 4 = 2(− 2)3 – 3(− 2) + 4 = 2(− 8) + 6 + 4 =−6 33 Combining Like Terms Like terms are terms that contain exactly the same variables raised to exactly the same powers. Warning! Only like terms can be combined through addition and subtraction. Example Combine like terms to simplify. x2y + xy – y + 10x2y – 2y + xy = x2y + 10x2y + xy + xy – y – 2y (Like terms are grouped together) = (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y = 11x2y + 2xy – 3y 34 § 12.4 Adding and Subtracting Polynomials Adding and Subtracting Polynomials Adding Polynomials Combine all the like terms. Subtracting Polynomials Change the signs of the terms of the polynomial being subtracted, and then combine all the like terms. 36 Adding and Subtracting Polynomials Example Add or subtract each of the following, as indicated. 1) (3x – 8) + (4x2 – 3x +3) = 3x – 8 + 4x2 – 3x + 3 = 4x2 + 3x – 3x – 8 + 3 = 4x2 – 5 2) 4 – (– y – 4) = 4 + y + 4 = y + 4 + 4 = y + 8 3) (– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7) = – a2 + 1 – a2 + 3 + 5a2 – 6a + 7 = – a2 – a2 + 5a2 – 6a + 1 + 3 + 7 = 3a2 – 6a + 11 37 Adding and Subtracting Polynomials In the previous examples, after discarding the parentheses, we would rearrange the terms so that like terms were next to each other in the expression. You can also use a vertical format in arranging your problem, so that like terms are aligned with each other vertically. 38