§ 5.2

Polynomial Functions and Adding and Subtracting

Polynomials

Martin-Gay, Beginning and Intermediate Algebra, 4ed 2

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, 4ed 3

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.

Polynomial Vocabulary

Martin-Gay, Beginning and Intermediate Algebra, 4ed 4

Monomial is a polynomial with one term.

Binomial is a polynomial with two terms.

Trinomial is a polynomial with three terms.

Types of Polynomials

Martin-Gay, Beginning and Intermediate Algebra, 4ed 5

Degree of a termTo 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.

Degrees

Martin-Gay, Beginning and Intermediate Algebra, 4ed 6

Like terms are terms that contain exactly the same variables raised to exactly the same powers.

Combine like terms to simplify.

x2y + xy – y + 10x2y – 2y + xy

Only like terms can be combined through addition and subtraction.

Warning!

11x2y + 2xy – 3y= (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y =

= x2y + 10x2y + xy + xy – y – 2y (Like terms are grouped together)

Combining Like Terms

Example:

Martin-Gay, Beginning and Intermediate Algebra, 4ed 7

Adding PolynomialsTo add polynomials, combine all the like terms.

Adding Polynomials

Add.

(3x – 8) + (4x2 – 3x +3)

= 4x2 + 3x – 3x – 8 + 3

= 4x2 – 5

= 3x – 8 + 4x2 – 3x + 3

Example:

Martin-Gay, Beginning and Intermediate Algebra, 4ed 8

Subtracting PolynomialsTo subtract two polynomials, change the signs of the terms of the polynomial being subtracted and then add.

Subtracting Polynomials

Example:Subtract.

= 3a2 – 6a + 11

4 – (– y – 4) = 4 + y + 4 = y + 4 + 4 = y + 8

(– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7)

= – a2 + 1 – a2 + 3 + 5a2 – 6a + 7

= – a2 – a2 + 5a2 – 6a + 1 + 3 + 7