Polynomial Zeros: Division, Remainders, and Factors

3.3 - 3.4 The Real Zeros of a Polynomial Function

Long Division of Polynomials

  • Purpose: Used when a polynomial divisor has more than one term.

  • Underlying Principle: Employs a repetitive procedure similar to whole number division:

    1. Divide

    2. Multiply

    3. Subtract

    4. Bring down the next term

  • Steps for Polynomial Long Division:

    1. Arrange Terms: Both the dividend and the divisor must have their terms arranged in descending powers of any variable. If any powers are missing, it's often helpful to include them with a coefficient of zero.

    2. Divide First Terms: Divide the first term of the dividend by the first term of the divisor. This result forms the first term of the quotient.

    3. Multiply: Multiply every term in the divisor by the first term you just found in the quotient. Write this product beneath the dividend, carefully aligning like terms.

    4. Subtract: Subtract the product from the dividend. Remember to distribute the negative sign to all terms of the product.

    5. Bring Down: Bring down the next term from the original dividend and place it next to the remainder obtained from the subtraction. This combined expression now becomes the new dividend.

    6. Repeat: Use this new expression as the dividend and repeat steps 2-5. Continue this process until the remainder can no longer be divided. This occurs when the degree (highest exponent on a variable) of the remainder is less than the degree of the divisor.

  • Example 1: Divide x^3 + 10x^2 + 21x - 5 by x^2 + 3x - 1.

  • Example 2: Divide 4 - 5x - x^2 + 6x^3 by 3x - 2. (Note: Rearrange to 6x^3 - x^2 - 5x + 4 by 3x - 2 first).

  • Example 4: Divide 6x^4 + 5x^3 + 3x - 5 by 3x^2 - 2x.

The Division Algorithm for Polynomials

  • Verification: Polynomial long division can be checked by multiplying the divisor by the quotient and then adding the remainder. This sum should reconstruct the original dividend.

  • Theorem 3 (Division Algorithm):

    • If f(x) and d(x) are polynomials, with d(x) \ne 0, and the degree of d(x) is less than or equal to the degree of f(x), then there exist unique polynomials q(x) (quotient) and r(x) (remainder) such that:
      f(x) = d(x)q(x) + r(x)

    • Remainder Condition: The remainder, r(x), must either be equal to 0 or its degree must be less than the degree of d(x).

    • Implication for Factors: If r(x) = 0, it means that d(x) divides evenly into f(x). In this case, both d(x) and q(x) are considered factors of f(x).

Synthetic Division

  • When Applicable: Synthetic division is a streamlined method specifically used to divide polynomials when the divisor is of the special form (x - c).

  • Steps for Synthetic Division:

    1. Arrange Polynomial: Write the dividend polynomial in descending powers. Crucially, include a coefficient of 0 for any missing terms (powers of x).

    2. Set Up: Write the value c (from the divisor x - c) in a box to the left. To its right, write down only the coefficients of the dividend polynomial.

    3. Bring Down: Bring the leading coefficient of the dividend directly down to the bottom row.

    4. Multiply: Multiply c by the number you just wrote on the bottom row. Write this product in the next column, in the second row.

    5. Add: Add the values in the new column (from the dividend coefficient and the product). Write the sum in the bottom row.

    6. Repeat: Continuously repeat steps 4 and 5 (multiply c by the latest bottom-row value, then add to the next column) until all columns containing coefficients are filled.

    7. Write Quotient and Remainder: The numbers in the final bottom row represent the coefficients of the quotient polynomial and the remainder.

      • The last value in the bottom row is the remainder.

      • The other numbers are the coefficients of the quotient. The degree of the first term of the quotient is one less than the degree of the first term of the original dividend.

  • Example 5: Use synthetic division to divide x^3 + 4x^2 - 5x + 5 by x - 3. (Here, c=3).

  • Example 6: Use synthetic division to divide 5x^3 + 6x + 8 by x + 2. (Note: x+2 is x - (-2), so c=-2. Also, include a 0 coefficient for the missing x^2 term: 5x^3 + 0x^2 + 6x + 8).

The Remainder Theorem

  • Theorem 7: If a polynomial f(x) is divided by (x - c), then the remainder obtained from this division is equal to the value of the function evaluated at c, i.e., f(c).

The Factor Theorem

  • Theorem 8: This theorem establishes a direct link between the zeros of a polynomial and its factors.

    1. If f(c) = 0: If, when a number c is substituted into a polynomial f(x), the result is 0 (meaning c is a zero of the polynomial), then (x - c) is a factor of f(x).

    2. If (x - c) is a factor: Conversely, if (x - c) is a factor of a polynomial f(x), then evaluating the polynomial at c will yield 0 (i.e., f(c) = 0), which means c is a zero of the polynomial.

  • Example 9: Solve the equation 2x^3 - 3x^2 - 11x + 6 = 0 given that 3 is a zero of f(x) = 2x^3 - 3x^2 - 11x + 6. Since 3 is a zero, by the Factor Theorem, (x - 3) must be a factor. We can use synthetic division with c=3 to find the other factors.

The Rational Zero Theorem

  • Theorem 10: This theorem helps in finding potential rational zeros of a polynomial.

    • Conditions: For a polynomial f(x) = anx^n + a{n-1}x^{n-1} + \dots + a1x + a0 with integer coefficients.

    • Rational Zero Form: If \frac{p}{q} (where \frac{p}{q} is a rational number reduced to lowest terms) is a rational zero of f(x):

      • p (the numerator) must be a factor of the constant term, a_0.

      • q (the denominator) must be a factor of the leading coefficient, a_n.

  • Strategy: This theorem provides a finite list of possible rational zeros to test. Once a rational zero is found (e.g., by synthetic division resulting in a zero remainder), the polynomial can be factored, simplifying the search for remaining zeros.

  • Example 11: List all possible rational zeros of f(x) = -x^4 + 3x^2 + 4.

    • Here, an = -1 and a0 = 4.

    • Factors of a_0 = 4 (which is p): \pm 1, \pm 2, \pm 4.

    • Factors of a_n = -1 (which is q): \pm 1.

    • Possible Rational Zeros \frac{p}{q}: \frac{\pm 1}{\pm 1}, \frac{\pm 2}{\pm 1}, \frac{\pm 4}{\pm 1} which simplifies to \pm 1, \pm 2, \pm 4.

  • Example 12: Find all zeros of f(x) = x^3 + 2x^2 - 5x - 6.

    • Use the Rational Zero Theorem to generate a list of possible rational zeros (factors of -6 over factors of 1).

    • Test these possibilities using synthetic division or direct substitution until a zero is found.

    • Once a zero is found, say c, then (x - c) is a factor, and the quotient is a polynomial of lower degree. Factor the quotient or apply the quadratic formula if it's a quadratic to find the remaining zeros.

  • Example 13: Find all zeros of f(x) = x^3 + 7x^2 + 11x - 3.

    • Similar process as Example 12. Find possible rational zeros (factors of -3 over factors of 1) and test them.