Unit 4 Polynomial/Rational Functions Remainder and Factor Theorems - - PowerPoint PPT Presentation

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:

• 6x3 + 17x2 − 104x + 60
• ÷ (2x − 5)

2x − 5

• 6x3 + 17x2 − 104x + 60
• 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:

• 6x3 + 17x2 − 104x + 60
• ÷ (2x − 5)

2x − 5

• 6x3 + 17x2 − 104x + 60
• W. Finch

DHS Math Dept Division 4 / 15

Introduction Long Synthetic Applications Summary

Example 2

Use long division to find the quotient. 4x3 − 9x − 3 x − 2 Note the zero place-holder in the dividend. x − 2

• 4x3 + 0x2 − 9x − 3
• W. Finch

DHS Math Dept Division 5 / 15

Introduction Long Synthetic Applications Summary

Example 2

Use long division to find the quotient. 4x3 − 9x − 3 x − 2 Note the zero place-holder in the dividend. x − 2

• 4x3 + 0x2 − 9x − 3
• W. Finch

DHS Math Dept Division 5 / 15

Introduction Long Synthetic Applications Summary

Example 3

Use long division to find the quotient.

• 6x4 − x3 − x2 + 9x − 3
• ÷
• x2 + x − 1
• x2 + x − 1
• 6x4 − x3 − x2 + 9x − 3
• W. Finch

DHS Math Dept Division 6 / 15

Introduction Long Synthetic Applications Summary

Example 3

Use long division to find the quotient.

• 6x4 − x3 − x2 + 9x − 3
• ÷
• x2 + x − 1
• x2 + x − 1
• 6x4 − x3 − x2 + 9x − 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

• f 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

• f 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

• f 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

• f 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

• f 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).

• ax3 + bx2 + cx + d
• ÷ (x − k)

k a b c d a + ka × (b + ka) Coeff of Quotient Remainder Basic Procedure

◮ Add down ◮ Multiply

diagonally

• 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).

• ax3 + bx2 + cx + d
• ÷ (x − k)

k a b c d a + ka × (b + ka) Coeff of Quotient Remainder Basic Procedure

◮ Add down ◮ Multiply

diagonally

• 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).

• ax3 + bx2 + cx + d
• ÷ (x − k)

k a b c d a + ka × (b + ka) Coeff of Quotient Remainder Basic Procedure

◮ Add down ◮ Multiply

diagonally

• 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).

• ax3 + bx2 + cx + d
• ÷ (x − k)

k a b c d a + ka × (b + ka) Coeff of Quotient Remainder Basic Procedure

◮ Add down ◮ Multiply

diagonally

• 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).

• ax3 + bx2 + cx + d
• ÷ (x − k)

k a b c d a + ka × (b + ka) Coeff of Quotient Remainder Basic Procedure

◮ Add down ◮ Multiply

diagonally

• 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).

• ax3 + bx2 + cx + d
• ÷ (x − k)

k a b c d a + ka × (b + ka) Coeff of Quotient Remainder Basic Procedure

◮ Add down ◮ Multiply

diagonally

• 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).

• ax3 + bx2 + cx + d
• ÷ (x − k)

k a b c d a + ka × (b + ka) Coeff of Quotient Remainder Basic Procedure

◮ Add down ◮ Multiply

diagonally

• W. Finch

DHS Math Dept Division 8 / 15

Introduction Long Synthetic Applications Summary

Example 4

Find the quotient using synthetic division.

• 3x3 − 5x2 + 9x + 10
• ÷ (x + 2)
• W. Finch

DHS Math Dept Division 9 / 15

Introduction Long Synthetic Applications Summary

Example 5

Find the quotient using synthetic division. 2x3 − 32x x − 4

• W. Finch

DHS Math Dept Division 10 / 15

Introduction Long Synthetic Applications Summary

Example 6

Find the quotient using synthetic division. 8x4 + 38x3 + 5x2 + 3x + 3 4x + 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

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

Introduction Long Synthetic Applications Summary

Example 7 – Illustration of Remainder Theorem

Given f (x) = 2x3 − 9x2 + 27 a) Find f (−2) by substitution. b) Find f (−2) using synthetic substitution.

• W. Finch

DHS Math Dept Division 13 / 15

Introduction Long Synthetic Applications Summary

Example 8 – Illustration of Factor Theorem

Use the factor theorem to determine if the binomials given are factors of f . Use the binomials that are factors to rewrite f in a factored form. f (x) = x4 − 11x3 + 43x2 − 69x + 36; (x − 1) and (x − 4)

• W. Finch

DHS Math Dept Division 14 / 15

Introduction Long Synthetic Applications Summary

What You Learned

You can now:

◮ 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. ◮ Do problems Chap 2.3 #1, 7, 9, 15, 19, 21, 27, 31, 39,

41, 45

• W. Finch

DHS Math Dept Division 15 / 15

Introduction Long Synthetic Applications Summary

What You Learned

You can now:

◮ 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. ◮ Do problems Chap 2.3 #1, 7, 9, 15, 19, 21, 27, 31, 39,

41, 45

• W. Finch

DHS Math Dept Division 15 / 15

Introduction Long Synthetic Applications Summary

What You Learned

You can now:

◮ 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. ◮ Do problems Chap 2.3 #1, 7, 9, 15, 19, 21, 27, 31, 39,

41, 45

• W. Finch

DHS Math Dept Division 15 / 15

Introduction Long Synthetic Applications Summary

What You Learned

You can now:

◮ 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. ◮ Do problems Chap 2.3 #1, 7, 9, 15, 19, 21, 27, 31, 39,

41, 45

• W. Finch

DHS Math Dept Division 15 / 15

Introduction Long Synthetic Applications Summary

What You Learned

You can now:

◮ 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. ◮ Do problems Chap 2.3 #1, 7, 9, 15, 19, 21, 27, 31, 39,

41, 45

• W. Finch

DHS Math Dept Division 15 / 15