171S4.3 Polynomial Division; The Remainder and Factor Theorems
October 26, 2010
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 4: Polynomial and Rational Functions
4.1 Polynomial Functions and Models
4.2 Graphing Polynomial Functions
4.3 Polynomial Division; The Remainder and Factor Theorems
4.4 Theorems about Zeros of Polynomial Functions
4.5 Rational Functions
4.6 Polynomial and Rational Inequalities
Click the globe to the left and visit SAS Curriculum Pathways for interactive programs on Polynomial Functions. Quick Launch 1213, 1245
Division and Factors
When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is 0, then the divisor is a factor of the dividend.
4.3 Polynomial Division; The Remainder and Factor Theorems
Perform long division with polynomials and determine whether one polynomial is a factor of another.
• Use synthetic division to divide a polynomial by x − c.
• Use the remainder theorem to find a function value f (c).
• Use the factor theorem to determine whether x − c is a factor of f (x).
See the following lesson in Course Documents of CourseCompass:
171Session4
171Session4 ( Package file )
This lesson is a brief discussion of and suggestions relative to studying Chapter 4.
Division and Factors continued
Divide:
Example: Divide to determine whether x + 3 and x − 1 are factors of Since the remainder is –64 ≠ 0, we know that x + 3 is not a factor.
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171S4.3 Polynomial Division; The Remainder and Factor Theorems
Division and Factors continued
Divide: Since the remainder is 0, we know that x − 1 is a factor.
The Remainder Theorem
If a number c is substituted for x in a polynomial f (x), then the result f (c) is the remainder that would be obtained by dividing f (x) by x − c. That is, if f (x) = (x − c) • Q(x) + R, then f (c) = R.
Synthetic division is a “collapsed” version of long division; only the coefficients of the terms are written.
October 26, 2010
Division of Polynomials
When dividing a polynomial P(x) by a divisor d(x), a polynomial Q(x) is the quotient and a polynomial R(x) is the remainder. The quotient must have degree less than that of the dividend, P(x). The remainder must be either 0 or have degree less than that of the divisor. P(x) = d(x) • Q(x) + R(x)
Dividend Divisor Quotient Remainder
Example
Use synthetic division to find the quotient and remainder.
Note: We must write a 0 for the missing term.
The quotient is – 4x4 – 7x3 – 8x2 – 14x – 28 and the remainder is –6.
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171S4.3 Polynomial Division; The Remainder and Factor Theorems
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The Factor Theorem
Example
Determine whether 4 is a zero of f(x), where f (x) = x3 − 6x2 + 11x − 6.
We use synthetic division and the remainder theorem to find f (4).
For a polynomial f (x), if f (c) = 0, then x − c is a factor of f (x).
Example: Let f (x) = x3 − 7x + 6. Factor f (x) and solve the equation f (x) = 0.
Solution: We look for linear factors of the form x − c. Let’s try x − 1:
f (4)
Since f (4) ≠ 0, the number 4 is not a zero of f (x).
Example continued
Since f (1) = 0, we know that x − 1 is one factor and the quotient x2 + x − 6 is another factor.
So, f (x) = (x − 1)(x + 3)(x − 2).
For f (x) = 0, we have x = − 3, 1, or 2. Each root has a multiplicity of one; thus, the graph crosses the x­axis at each of the x­intercepts (­3,0), (1,0), and (2,0).
329/2. Use long division to determine whether (a) x + 5 is a factor of h(x) = x3 ­ x2 ­ 17x ­ 15.
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171S4.3 Polynomial Division; The Remainder and Factor Theorems
329/10. Use long division to find the quotient Q(x) and the remainder R(x). Express P(x) = d(x)Q(x) + R(x).
P(x) = x4 + 6x3; d(x) = x ­ 1
329/22. Use synthetic division to find the quotient and remainder. (3x4 ­ 2x2 + 2) ÷ (x ­ 1/4)
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329/14. Use synthetic division to find the quotient and remainder. (x3 ­ 3x + 10) ÷ (x ­ 2)
329/26. Use synthetic division to find the function values. Check with calculator. f(x) = 2x4 + x2 ­ 10x + 1; f(­10) = 4
171S4.3 Polynomial Division; The Remainder and Factor Theorems
329/30. Use synthetic division to find the function value of f(2 + 3i) when f(x) = x5 + 32. Check with calculator. October 26, 2010
330/35. Using synthetic division, determine whether the numbers are zeros of the polynomial function g(x) = x3 ­ 4x2 + 4x ­ 16; i, ­2i
Since g(­2i) = 0, x = ­2i is a zero or root of g(x).
330/48. Factor the polynomial function. Then solve f(x) = 0.
f(x) = x4 + 11x3 + 41x2 + 61x + 30
330/44. Factor the polynomial function. Then solve f(x) = 0.
f(x) = x3 ­ 3x2 ­ 10x + 24
By using synthetic division, we find f(2) = 0. So, x ­ 2 is a factor. Also, Q(x) = x2 ­ x ­ 12 is another factor. This can be factored as x2 ­ x ­ 12 = (x ­ 4)(x + 3). Thus, f(x) can be written as x3 ­ 3x2 ­ 10x + 24 = (x ­ 2)(x ­ 4)(x + 3) = 0.
So, f(x) = 0 when x = 2, x = 4, or x = ­3.
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171S4.3 Polynomial Division; The Remainder and Factor Theorems
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330/50. Sketch the graph of the function. Use synthetic division and the remainder theorem to find zeros.
f(x) = x4 + x3 ­ 3x2 ­ 5x ­ 2
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