WARM UP Simplify 1. 2.

DIVISION OF POLYNOMIALS

OBJECTIVES  Divide a polynomial by a monomial.  Divide two polynomials when the divisor is not a monomial.  Solve problems involving division of polynomials

DIVIDING POLYNOMIALS BY MONOMIALS  Remember that rational expressions indicate division. In some cases, it is useful to carry out that division.  Division by a monomial can be done by first writing a rational expression.

EXAMPLE 1 Divide by 4x Writing a rational expression Dividing each term by 4x Simplifying each term

TRY THIS… Divide

EXAMPLE 2 Divide by Writing a rational expression Dividing each term by 4x Simplifying each term DIVIDING A POLYNOMIAL BY A MONOMIAL To divide a polynomial by a monomial, divide each term by the monomial.

TRY THIS… Divide

DIVIDING TWO POLYNOMIALS  When the divisor is not a monomial, we use a procedure very much like long division.  In long-division notation, the dividend is inside the division symbol, the divisor is to the left of it, and the quotient is above it.  Note that as with any fraction, the rational fraction representing the division may also be called a quotient.

EXAMPLE 3 Divide by x + 3 Writing in long-division notation Dividing the first term of the dividend by the first term of the divisor Multiplying the divisor by x Subtracting We now “bring down” 8, the next term of the dividend. Dividing the first term of the dividend by the first term of the divisor The quotient is x + 2 and the remainder is 2.

CHECK YOUR SOLUTION To check, we multiply the quotient by the divisor and add the remainder to see if we get the dividend. Quotient  Divisor + Remainder = Dividend (x + 2)  (x + 3) + 2 = The answer checks

ARRANGING POLYNOMIALS FOR DIVIDING  Always remember to arrange polynomials in descending order and to leave space for missing terms in the dividend (or write them with 0 coefficients). Example 4: Divide by Writing zero coefficients for missing terms The quotient is

EXAMPLE 5 Example 4: Divide by Bring down the “9” Multiply x – 2 by -10 = - 10x + 20 Bring down the “0” Multiply x – 2 by 5x = 5x -10x The quotient is and the remainder is -25. This can be written as, R -25 or: The expression in red is the remainder over the divisor. 4

TRY THIS… Divide and check 1. 2

DIVIDING POLYNOMIALS WITH REMAINDERS  When dividing, we continue until the degree of the remainder is less than the degree of the divisor.  The answer can be written as a quotient and remainder, or as a polynomial plus a rational expression.

EXAMPLE 6 Example 4: Divide by Leaving space for the missing term. The degree of the remainder is less than the degree of the divisor, so we are finished. The quotient is and the remainder is. Checking: The answer checks. 3

TRY THIS… Divide and check

CH. 6.4 HOMEWORK Textbook pg. 262 #2, 4, 8, 16, 20 & 26.