5 2 Polynomial Functions and Adding and Subtracting

§ 5. 2 Polynomial Functions and Adding and Subtracting 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. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 2

Polynomial Vocabulary In the polynomial 7 x 5 + x 2 y 2 – 4 xy + 7, there are 4 terms: 7 x 5, x 2 y 2, – 4 xy and 7. The coefficient of term 7 x 5 is 7, of term x 2 y 2 is 1, of term – 4 xy is – 4 and of term 7 is 7. 7 is a constant term. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 3

Types of Polynomials Monomial is a polynomial with one term. Binomial is a polynomial with two terms. Trinomial is a polynomial with three terms. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 4

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 5 a 4 b 3 c is 8 (remember that c can be written as c 1). Degree of a polynomial To find the degree, take the largest degree of any term of the polynomial. Degree of 9 x 3 – 4 x 2 + 7 is 3. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 5

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. x 2 y + xy – y + 10 x 2 y – 2 y + xy = x 2 y + 10 x 2 y + xy – 2 y (Like terms are grouped together) = (1 + 10)x 2 y + (1 + 1)xy + (– 1 – 2)y = 11 x 2 y + 2 xy – 3 y Martin-Gay, Beginning and Intermediate Algebra, 4 ed 6

Adding Polynomials To add polynomials, combine all the like terms. Example: Add. (3 x – 8) + (4 x 2 – 3 x +3) = 3 x – 8 + 4 x 2 – 3 x + 3 = 4 x 2 + 3 x – 8 + 3 = 4 x 2 – 5 Martin-Gay, Beginning and Intermediate Algebra, 4 ed 7

Subtracting Polynomials To subtract two polynomials, change the signs of the terms of the polynomial being subtracted and then add. Example: Subtract. 4 – (– y – 4) = 4 + y + 4 = y + 8 (– a 2 + 1) – (a 2 – 3) + (5 a 2 – 6 a + 7) = – a 2 + 1 – a 2 + 3 + 5 a 2 – 6 a + 7 = – a 2 + 5 a 2 – 6 a + 1 + 3 + 7 = 3 a 2 – 6 a + 11 Martin-Gay, Beginning and Intermediate Algebra, 4 ed 8