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Unit 4 Polynomial/Rational Functions Remainder and Factor Theorems (Chap 2.3) William (Bill) Finch Mathematics Department Denton High School Introduction Long Synthetic Applications Summary Lesson Goals When you have completed this

Unit 4 – Polynomial/Rational Functions Remainder and Factor Theorems (Chap 2.3) William (Bill) Finch Mathematics Department Denton High School
Introduction Long Synthetic Applications Summary Lesson Goals When you have completed this lesson you will: ◮ Use long division to divide polynomials. ◮ Use synthetic division to divide a polynomial by a linear divisor of the form ( x − a ) . ◮ Apply the Remainder Theorem and Factor Theorem. W. Finch DHS Math Dept Division 2 / 15
Introduction Long Synthetic Applications Summary Lesson Goals When you have completed this lesson you will: ◮ Use long division to divide polynomials. ◮ Use synthetic division to divide a polynomial by a linear divisor of the form ( x − a ) . ◮ Apply the Remainder Theorem and Factor Theorem. W. Finch DHS Math Dept Division 2 / 15
Introduction Long Synthetic Applications Summary Lesson Goals When you have completed this lesson you will: ◮ Use long division to divide polynomials. ◮ Use synthetic division to divide a polynomial by a linear divisor of the form ( x − a ) . ◮ Apply the Remainder Theorem and Factor Theorem. W. Finch DHS Math Dept Division 2 / 15
Introduction Long Synthetic Applications Summary Lesson Goals When you have completed this lesson you will: ◮ Use long division to divide polynomials. ◮ Use synthetic division to divide a polynomial by a linear divisor of the form ( x − a ) . ◮ Apply the Remainder Theorem and Factor Theorem. W. Finch DHS Math Dept Division 2 / 15
Introduction Long Synthetic Applications Summary Long Division Recall the long division process from elementary school for 3285 ÷ 21 : � 21 3285 W. Finch DHS Math Dept Division 3 / 15
Introduction Long Synthetic Applications Summary Example 1 You can also use long division for dividing polynomials such as: 6 x 3 + 17 x 2 − 104 x + 60 � � ÷ (2 x − 5) 6 x 3 + 17 x 2 − 104 x + 60 � 2 x − 5 W. Finch DHS Math Dept Division 4 / 15
Introduction Long Synthetic Applications Summary Example 1 You can also use long division for dividing polynomials such as: 6 x 3 + 17 x 2 − 104 x + 60 � � ÷ (2 x − 5) 6 x 3 + 17 x 2 − 104 x + 60 � 2 x − 5 W. Finch DHS Math Dept Division 4 / 15
Introduction Long Synthetic Applications Summary Example 2 Use long division to find the quotient. 4 x 3 − 9 x − 3 x − 2 Note the zero place-holder in the dividend. 4 x 3 + 0 x 2 − 9 x − 3 � x − 2 W. Finch DHS Math Dept Division 5 / 15
Introduction Long Synthetic Applications Summary Example 2 Use long division to find the quotient. 4 x 3 − 9 x − 3 x − 2 Note the zero place-holder in the dividend. 4 x 3 + 0 x 2 − 9 x − 3 � x − 2 W. Finch DHS Math Dept Division 5 / 15
Introduction Long Synthetic Applications Summary Example 3 Use long division to find the quotient. 6 x 4 − x 3 − x 2 + 9 x − 3 x 2 + x − 1 � � � � ÷ x 2 + x − 1 6 x 4 − x 3 − x 2 + 9 x − 3 � W. Finch DHS Math Dept Division 6 / 15
Introduction Long Synthetic Applications Summary Example 3 Use long division to find the quotient. 6 x 4 − x 3 − x 2 + 9 x − 3 x 2 + x − 1 � � � � ÷ x 2 + x − 1 6 x 4 − x 3 − x 2 + 9 x − 3 � W. Finch DHS Math Dept Division 6 / 15
Introduction Long Synthetic Applications Summary The Division Algorithm If f ( x ) and d ( x ) are polynomials ( d ( x ) � = 0), and the degree of d ( x ) is less than or equal to the degree of f ( x ), then there are unique polynomials q ( x ) and r ( x ) such that f ( x ) = d ( x ) q ( x ) + r ( x ) where ◮ f ( x ) is the dividend ◮ d ( x ) is the divisor ◮ q ( x ) is the quotient ◮ r ( x ) is the remainder W. Finch DHS Math Dept Division 7 / 15
Introduction Long Synthetic Applications Summary The Division Algorithm If f ( x ) and d ( x ) are polynomials ( d ( x ) � = 0), and the degree of d ( x ) is less than or equal to the degree of f ( x ), then there are unique polynomials q ( x ) and r ( x ) such that f ( x ) = d ( x ) q ( x ) + r ( x ) where ◮ f ( x ) is the dividend ◮ d ( x ) is the divisor ◮ q ( x ) is the quotient ◮ r ( x ) is the remainder W. Finch DHS Math Dept Division 7 / 15
Introduction Long Synthetic Applications Summary The Division Algorithm If f ( x ) and d ( x ) are polynomials ( d ( x ) � = 0), and the degree of d ( x ) is less than or equal to the degree of f ( x ), then there are unique polynomials q ( x ) and r ( x ) such that f ( x ) = d ( x ) q ( x ) + r ( x ) where ◮ f ( x ) is the dividend ◮ d ( x ) is the divisor ◮ q ( x ) is the quotient ◮ r ( x ) is the remainder W. Finch DHS Math Dept Division 7 / 15
Introduction Long Synthetic Applications Summary The Division Algorithm If f ( x ) and d ( x ) are polynomials ( d ( x ) � = 0), and the degree of d ( x ) is less than or equal to the degree of f ( x ), then there are unique polynomials q ( x ) and r ( x ) such that f ( x ) = d ( x ) q ( x ) + r ( x ) where ◮ f ( x ) is the dividend ◮ d ( x ) is the divisor ◮ q ( x ) is the quotient ◮ r ( x ) is the remainder W. Finch DHS Math Dept Division 7 / 15
Introduction Long Synthetic Applications Summary The Division Algorithm If f ( x ) and d ( x ) are polynomials ( d ( x ) � = 0), and the degree of d ( x ) is less than or equal to the degree of f ( x ), then there are unique polynomials q ( x ) and r ( x ) such that f ( x ) = d ( x ) q ( x ) + r ( x ) where ◮ f ( x ) is the dividend ◮ d ( x ) is the divisor ◮ q ( x ) is the quotient ◮ r ( x ) is the remainder W. Finch DHS Math Dept Division 7 / 15
Introduction Long Synthetic Applications Summary Synthetic Division Synthetic division is a useful shortcut for long division when the divisor is of the form ( x − k ). ax 3 + bx 2 + cx + d � � ÷ ( x − k ) Basic Procedure k a b c d ◮ Add down + ka × ◮ Multiply diagonally Remainder ( b + ka ) a Coeff of Quotient W. Finch DHS Math Dept Division 8 / 15
Introduction Long Synthetic Applications Summary Synthetic Division Synthetic division is a useful shortcut for long division when the divisor is of the form ( x − k ). ax 3 + bx 2 + cx + d � � ÷ ( x − k ) Basic Procedure k a b c d ◮ Add down + ka × ◮ Multiply diagonally Remainder ( b + ka ) a Coeff of Quotient W. Finch DHS Math Dept Division 8 / 15
Introduction Long Synthetic Applications Summary Synthetic Division Synthetic division is a useful shortcut for long division when the divisor is of the form ( x − k ). ax 3 + bx 2 + cx + d � � ÷ ( x − k ) Basic Procedure k a b c d ◮ Add down + ka × ◮ Multiply diagonally Remainder ( b + ka ) a Coeff of Quotient W. Finch DHS Math Dept Division 8 / 15
Introduction Long Synthetic Applications Summary Synthetic Division Synthetic division is a useful shortcut for long division when the divisor is of the form ( x − k ). ax 3 + bx 2 + cx + d � � ÷ ( x − k ) Basic Procedure k a b c d ◮ Add down + ka × ◮ Multiply diagonally Remainder ( b + ka ) a Coeff of Quotient W. Finch DHS Math Dept Division 8 / 15
Introduction Long Synthetic Applications Summary Synthetic Division Synthetic division is a useful shortcut for long division when the divisor is of the form ( x − k ). ax 3 + bx 2 + cx + d � � ÷ ( x − k ) Basic Procedure k a b c d ◮ Add down + ka × ◮ Multiply diagonally Remainder ( b + ka ) a Coeff of Quotient W. Finch DHS Math Dept Division 8 / 15
Introduction Long Synthetic Applications Summary Synthetic Division Synthetic division is a useful shortcut for long division when the divisor is of the form ( x − k ). ax 3 + bx 2 + cx + d � � ÷ ( x − k ) Basic Procedure k a b c d ◮ Add down + ka × ◮ Multiply diagonally Remainder ( b + ka ) a Coeff of Quotient W. Finch DHS Math Dept Division 8 / 15
Introduction Long Synthetic Applications Summary Synthetic Division Synthetic division is a useful shortcut for long division when the divisor is of the form ( x − k ). ax 3 + bx 2 + cx + d � � ÷ ( x − k ) Basic Procedure k a b c d ◮ Add down + ka × ◮ Multiply diagonally Remainder ( b + ka ) a Coeff of Quotient W. Finch DHS Math Dept Division 8 / 15
Introduction Long Synthetic Applications Summary Example 4 Find the quotient using synthetic division. 3 x 3 − 5 x 2 + 9 x + 10 � � ÷ ( x + 2) W. Finch DHS Math Dept Division 9 / 15
Introduction Long Synthetic Applications Summary Example 5 Find the quotient using synthetic division. 2 x 3 − 32 x x − 4 W. Finch DHS Math Dept Division 10 / 15
Introduction Long Synthetic Applications Summary Example 6 Find the quotient using synthetic division. 8 x 4 + 38 x 3 + 5 x 2 + 3 x + 3 4 x + 1 W. Finch DHS Math Dept Division 11 / 15
Introduction Long Synthetic Applications Summary Remainder Theorem and Factor Theorem Remainder Theorem If a polynomial f ( x ) is divided by ( x − k ), the remainder is r = f ( k ). Factor Theorem A polynomial f ( x ) has a factor ( x − k ) if and only if f ( k ) = 0. W. Finch DHS Math Dept Division 12 / 15

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