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Unit 6: Quadratic and Polynomial Functions

Topic 6

Zeros, Factors, and Graphs of Polynomials

A zero (or root) of a polynomial function is an xx-value that makes the function equal to zero:
f(x)=0.f(x) = 0.
Each zero corresponds to an x-intercept on the graph of the polynomial.
A factor is a binomial or term that divides the polynomial evenly. If (x−r)(x - r) is a factor of f(x)f(x), then rr is a zero of the function. This is known as the Factor Theorem.
f(r)=0⇔(x−r) is a factor of f(x).f(r) = 0 \quad \Leftrightarrow \quad (x - r) \text{ is a factor of } f(x).
The graph of a polynomial touches or crosses the x-axis at its zeros, depending on their multiplicity:
  • If a zero has odd multiplicity, the graph crosses the x-axis.
  • If a zero has even multiplicity, the graph touches the x-axis and turns around.
End Behavior The leading term (the term with the highest power of xx) determines how the graph behaves at the ends:
f(x)=anxn+…f(x) = a_nx^n + \dots
{an>0,n even: both ends upan<0,n even: both ends downan>0,n odd: left down, right upan<0,n odd: left up, right down\begin{cases} a_n > 0, n \text{ even: both ends up} \\ a_n < 0, n \text{ even: both ends down} \\ a_n > 0, n \text{ odd: left down, right up} \\ a_n < 0, n \text{ odd: left up, right down} \end{cases}

Core Skills

  • Identify zeros from factored forms of polynomials.
  • Determine multiplicity and predict graph behavior at each zero.
  • Use the Factor Theorem to test if a given value is a zero.
  • Sketch graphs based on zeros, multiplicities, and end behavior.
  • Connect factored form, standard form, and graphical interpretations.

Example 1: Identifying Zeros from Factored Form

Let f(x)=(x−2)(x+1)(x−4)f(x) = (x - 2)(x + 1)(x - 4). Step 1: Set each factor equal to zero:
x−2=0⇒x=2,x+1=0⇒x=−1,x−4=0⇒x=4.x - 2 = 0 \Rightarrow x = 2, \quad x + 1 = 0 \Rightarrow x = -1, \quad x - 4 = 0 \Rightarrow x = 4.
Zeros: −1,2,4.-1, 2, 4.

Example 2: Multiplicity of Zeros

Let f(x)=(x−3)2(x+1).f(x) = (x - 3)^2(x + 1). Zeros: x=3x = 3 (multiplicity 2), x=−1x = -1 (multiplicity 1). At x=3x = 3, the graph touches and turns around (even multiplicity). At x=−1x = -1, the graph crosses (odd multiplicity).

Example 3: Testing for Factors (Factor Theorem)

Let f(x)=x3−6x2+11x−6.f(x) = x^3 - 6x^2 + 11x - 6. Test if x−1x - 1 is a factor. Substitute x=1x = 1:
f(1)=1−6+11−6=0.f(1) = 1 - 6 + 11 - 6 = 0.
Since f(1)=0,f(1) = 0, (x−1)(x - 1) is a factor. Similarly, test x=2x = 2: f(2)=8−24+22−6=0.f(2) = 8 - 24 + 22 - 6 = 0. Also a factor. x=3:  f(3)=27−54+33−6=0.x = 3:\; f(3) = 27 - 54 + 33 - 6 = 0. All three are zeros. Factored form: f(x)=(x−1)(x−2)(x−3)\boxed{f(x) = (x - 1)(x - 2)(x - 3)}

Example 4: End Behavior and Degree

Determine the end behavior of:
f(x)=−2x4+5x3−x+7.f(x) = -2x^4 + 5x^3 - x + 7.
Leading term is −2x4.-2x^4. Since a=−2<0a = -2 < 0 and degree is even, both ends point downward.

Example 5: Building a Polynomial from Given Zeros

Find the polynomial with zeros x=−2,1,3.x = -2, 1, 3.
f(x)=(x+2)(x−1)(x−3).f(x) = (x + 2)(x - 1)(x - 3).
Expand:
(x+2)(x2−4x+3)=x3−4x2+3x+2x2−8x+6=x3−2x2−5x+6.(x + 2)(x^2 - 4x + 3) = x^3 - 4x^2 + 3x + 2x^2 - 8x + 6 = x^3 - 2x^2 - 5x + 6.
f(x)=x3−2x2−5x+6\boxed{f(x) = x^3 - 2x^2 - 5x + 6}

Key Takeaways

  • Zeros are x-values where f(x)=0.f(x) = 0.
  • If (x−r)(x - r) is a factor, then rr is a zero (Factor Theorem).
  • Even multiplicity → graph touches; odd multiplicity → graph crosses.
  • Leading term determines end behavior.
  • Factored form reveals zeros and structure quickly for sketching.