2.3Real Zeros of Polynomials Notes

Division Algorithm (Theorem 2.8)

• Key idea: polynomial long division works like integer division.
• If p(x) and d(x) are nonzero polynomials with deg(p) ≥ deg(d), there exist unique polynomials q(x) and r(x) such that
  $p(x) = d(x)\,q(x) + r(x)$
  where either $r(x) = 0$ or \deg r < \deg d.
• Interpretation:
  • q(x) is the quotient.
  • r(x) is the remainder.
  • The remainder must have strictly smaller degree than the divisor, unless it is zero.
• Intuition/metaphor: factoring p into a multiple of d plus a leftovers remainder, much like distributing apples into boxes with possibly a small leftover.
• Consequences:
  • The division is always possible for nonzero polynomials and the quotient and remainder are unique.
  • If the divisor degree is 1, the remainder is a constant (the Remainder Theorem is a corollary).

Example 1 (as given in transcript): Perform the indicated division

• (a) $(4y^{4} + 3y^{2} + 1) ÷ (2y^{2} - y + 1)$
  • We find a quotient and remainder such that
    $4y^{4} + 3y^{2} + 1 = (2y^{2} - y + 1)\,q(y) + r(y)$
    with \deg r < \deg(2y^{2} - y + 1) = 2.
  • Result (calculation): quotient $q(y) = 2y^{2} + y + 1$ and remainder $r(y) = 0$, so
    $4y^{4} + 3y^{2} + 1 = (2y^{2} - y + 1)(2y^{2} + y + 1).$
• (b) $(4z^{4} - 12z^{3} - 60z^{2} + 28z + 56) ÷ (z - 6)$
  • Here deg(divisor) = 1, so the remainder is a constant: r = p(6).
  • By synthetic division (or long division) the quotient is $q(z) = 4z^{3} + 12z^{2} + 12z + 100$ and the remainder is $r = 656$.
  • Therefore
    $4z^{4} - 12z^{3} - 60z^{2} + 28z + 56 = (z - 6)\bigl(4z^{3} + 12z^{2} + 12z + 100\bigr) + 656.$
• Takeaway: to check division results, you can verify by multiplying the divisor by the quotient and adding the remainder to recover the original polynomial.

The Remainder Theorem (Theorem 2.9)

• Statement: If p(x) is a polynomial of degree at least 1 and c is a real number, then the remainder of p(x) divided by (x − c) is p(c).
• Algebraic version:
  $p(x) = (x - c)\,q(x) + p(c).$
• Immediate corollary: for a linear divisor (x − c), the remainder is a constant p(c).
• Significance: evaluating p at c gives the remainder when dividing by (x − c); this provides a quick test for whether c is a root (if p(c) = 0, remainder is 0).

The Factor Theorem (Theorem 2.10)

• Statement: For a nonzero polynomial p, the real number c is a zero of p (i.e., p(c) = 0) if and only if (x − c) is a factor of p.
• Equivalently:
  $p(c) = 0 \iff (x - c) ext{ is a factor of } p(x).$
• Practical use: knowing a zero lets you factor out (x − c) and reduce the polynomial degree to find remaining zeros.

Example 2: Given a zero x = 4 of p(x) = 2x^{3} - 17x^{2} + 18x + 72, identify all remaining zeros

• Given: x = 4 is a zero, so by the Factor Theorem, (x − 4) is a factor.
• Divide p(x) by (x − 4) to obtain the quotient q(x): use synthetic division with root 4 on coefficients [2, -17, 18, 72].
  • Synthetic division result: quotient coefficients are [2, -9, -18], remainder 0.
  • Therefore
    $p(x) = (x - 4)\bigl(2x^{2} - 9x - 18\bigr).$
• Factor the quadratic: $2x^{2} - 9x - 18 = (2x + 3)(x - 6).$
• Complete factorization:
  $p(x) = (x - 4)(2x + 3)(x - 6).$
• Zeros of p(x):
  • $$x = 4,\