171S4.3 Polynomial Division; The Remainder and Factor Theorems
March 24, 2011
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
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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.
March 24, 2011
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
Example
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The Factor Theorem
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).
329/2. Use long division to determine whether (a) x + 5 is a factor of h(x) = x3 ­ x2 ­ 17x ­ 15.
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).
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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)
March 24, 2011
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. March 24, 2011
330/35. Using synthetic division, determine whether the numbers are zeros of the polynomial function g(x) = x3 ­ 4x2 + 4x ­ 16; i, ­2i
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
330/44. Factor the polynomial function. Then solve f(x) = 0.
f(x) = x3 ­ 3x2 ­ 10x + 24
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