This document discusses polynomials and partial fractions. It covers arithmetic operations on polynomials, identities involving polynomials, and dividing polynomials. Specifically, it teaches how to: 1) Perform addition, subtraction, and multiplication on polynomial expressions. 2) Identify polynomial equations and identities based on their properties. 3) Use the long division process to divide one polynomial by another.

1. Polynomials and Partial Fractions Objectives In this lesson, you will learn how to perform arithmetic operations on polynomials. 4.1 Polynomials

2. The following expressions are polynomials. We also use this form of function notation to denote a polynomial. 7 is the coefficient of x 4 and ½ is the coefficient of x 2 . A polynomial in a variable x, is a sum of terms, each of the form ax n , where a is a constant and n is a non-negative integer. Polynomials and Partial Fractions

3. Substitute for x in P( x ). Combine the two polynomial functions. If P( x ) = x 2 + x + 1 and Q( x ) = 2 x 2 – 3 x + 2, find Polynomials and Partial Fractions Example ( a ) P(3), ( b ) P( x ) + 2Q( x ).

4. Polynomials and Partial Fractions In this lesson, you will learn how to find unknown constants in a polynomial identity. 4.2 Identities Objectives

5. An expression involving polynomials that can be solved to find a specific value for x , is an equation . This is always true, so, it is an identity . We have solved for x, so, this is an equation. Equations and Identities Polynomials and Partial Fractions An expression involving polynomials that is true for all values of x is an identity .

6. If x = 2, then ( x – 2) = 0. The coefficients of x 2 could be used too. Find the values of a and b in the following identity. Polynomials and Partial Fractions Let x = 2. Equate the coefficients of x. Check the results graphically. Example

7. If x = 1, then ( x – 1) = 0. Find the values of A , B and C in the following identity. Polynomials and Partial Fractions Let x = 1. Let x = 2. If x = 2, then ( x – 2) = 0. Equate the coefficients of x 3 . Example

8. Polynomials and Partial Fractions In this lesson, you will learn how to divide one polynomial by another. 4.3 Dividing Polynomials Objectives

9. Subtract 1 × 8 from 13. Divide 13 by 8. A reminder about long division of integers. Polynomials and Partial Fractions Bring the 2 down. Divide 52 by 8. Subtract 6 × 8 from 52. Bring the 7 down. Divide 47 by 8. Subtract 5 × 8 from 47. divisor dividend quotient remainder

10. We will now apply the same process to polynomials. For any division, Polynomials and Partial Fractions dividend = divisor × quotient + remainder or dividend ÷ divisor = quotient +

11. Subtract x 3 × ( x – 1) from x 4 + 3 x 3 . Divide x 4 by x. This is the same method as long division with integers. Polynomials and Partial Fractions Bring the – 2 x 2 down. Divide 4 x 3 by x. Subtract 4 x 2 × ( x – 1) from 4 x 3 – 2 x 2 . Bring the x down. Divide 2 x 2 by x. Subtract 2 x × ( x – 1) from 2 x 2 + x. divisor dividend quotient remainder Divide 3 x by x. Subtract 3 × ( x – 1) from 3 x – 7. Bring the – 7 down.

12. The following identity is always true Polynomials and Partial Fractions dividend = divisor × quotient + remainder Therefore + remainder quotient divisor dividend

13. Example Divide . Subtract 2 x 2 × ( x – 2) from 2 x 3 – 4 x 2 . Divide 2 x 3 by x. Polynomials and Partial Fractions Bring the x down. Divide 0 by x. There is no x term. Bring the – 2 down. Divide x by x. Subtract 1 × ( x – 2) from x – 2.

14. Example Divide . Subtract 4 x 2 × ( x 2 – x + 2) from 4 x 4 – 5 x 3 + x 2 Divide 4 x 4 by x 2 . Polynomials and Partial Fractions No term to bring down. Divide – x 3 by x 2 . Subtract – x × ( x 2 – x + 2) from – x 3 – 7 x 2 . Bring the – 2 down. Subtract – 8 × ( x 2 – x + 2) from – 8 x 2 +8 x – 2. Divide –8 x 2 by x 2 .