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- 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 .

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