Polynomial Inequalities

What is a Polynomial Inequality

Let f(x) be a polynomial. A polynomial inequality is an inequality that can be written in the form

LaTeX: f\left(x\right)<0.f(x)<0.

The symbol < can be replaced with >, ≥, or ≤ .

How Do You Solve a Polynomial Inequality?

  1. Rewrite the equation or inequality, if necessary, so that an expression is on one side with 0 on the other side.

  2. Set the expression of the equation or inequality equal to ƒ(x), and graph the related function.

  3. Use the graph of ƒ(x) to determine solutions as follows.

    1. The real solutions of ƒ(x) = 0 are the x-values of the
      x-intercepts of the graph. These are the zeros of ƒ(x).

    2. The real solutions of ƒ(x) < 0 are the x-values for which the graph lies below the x-axis.

    3. The real solutions of ƒ(x) > 0 are the x-values for which the graph lies above the x-axis.

Examples

Use the graph to solve each equation or inequality

  1. LaTeX: -3\left(x-1\right)\left(x+4\right)=0−3(x−1)(x+4)=0
  2. LaTeX: -3\left(x-1\right)\left(x+4\right)<0−3(x−1)(x+4)<0
  3. LaTeX: -3\left(x-1\right)\left(x+4\right)\ge0−3(x−1)(x+4)≥0

Solutions

  1. LaTeX: -3\left(x-1\right)\left(x+4\right)=0−3(x−1)(x+4)=0
    Graph for Example 1 on Rational Inequalities.PNG

    The solutions of the equation are the x-values of the x-intercepts of the graph of ƒ(x), –4 and 1.

    Therefore, the solution set is {–4, 1}.

  2. LaTeX: -3\left(x-1\right)\left(x+4\right)<0−3(x−1)(x+4)<0
    Graph for Example 2 on Rational Inequalities.PNG

    The solutions of the inequality includes the x-values for which the graph of ƒ(x) lies below the x-axis. The solution set is LaTeX: \left(-\infty,-4\right)\cup\left(1,\infty\right).(−∞,−4)∪(1,∞).

  3. LaTeX: -3\left(x-1\right)\left(x+4\right)\ge0−3(x−1)(x+4)≥0

    Graph for Example 3 on Rational Inequalities-1.PNG

    The solutions of the inequality includes the x-values for which the graph of ƒ(x) lies above the x-axis. The solution set is [-4, 1].

Examples

Solve each equation or inequality.

  1. LaTeX: 5\left(x-1\right)^2\left(x-3\right)=05(x−1)2(x−3)=0
  2. LaTeX: 5\left(x-1\right)^2\left(x-3\right)<05(x−1)2(x−3)<0
  3. LaTeX: 5\left(x-1\right)^2\left(x-3\right)\ge05(x−1)2(x−3)≥0
  4. LaTeX: 5\left(x-1\right)^2\left(x-3\right)\le05(x−1)2(x−3)≤0

Solutions

Graph the function. The zeros of the function are 1 (multiplicity 2) and 3. Multiplying the factors and identifying the term of the greatest degree show that the dominating term is LaTeX: 5x^3.\:5x3.The y-intercept is (0, –15). Using the appropriate end behavior, draw a smooth curve that crosses the x-axis at 3 and touches the x-axis at 1, then turns and changes direction. The graph also passes through the y-intercept (0, –15).The graph is below.

Graph of function with a leading term of 5 x cubed.PNG

  1. LaTeX: 5\left(x-1\right)^2\left(x-3\right)=05(x−1)2(x−3)=0

    f(x) = 0 for the x-values of the x-intercepts of the graph that are the zeros of f(x). The solution set is {1, 3}.

  2. LaTeX: 5\left(x-1\right)^2\left(x-3\right)<05(x−1)2(x−3)<0

    The solution lies below the x-axis for x < 1 and 1 < x < 3. The solution set is LaTeX: \left(-\infty,1\right)\cup\left(1,3\right).(−∞,1)∪(1,3).

  3. LaTeX: 5\left(x-1\right)^2\left(x-3\right)\ge05(x−1)2(x−3)≥0

    The solution is where the graph lies on or above the x-axis. The solution set is LaTeX: \lbrace1\rbrace\cup{1}∪[3,LaTeX: \infty∞).

  4. LaTeX: 5\left(x-1\right)^2\left(x-3\right)\le05(x−1)2(x−3)≤0

    The solution lies on or below the x-axis. The solution set is (LaTeX: -\infty,3−∞,3].

Example

Solve LaTeX: 4x^4-6x^3-3x^2+x\ge2x^3-6x+2.4x4−6x3−3x2+x≥2x3−6x+2.

Solution

Add and subtract to get 0 on one side, then combine like terms:

LaTeX: 4x^4-8x^3-3x^2+7x-2\ge04x4−8x3−3x2+7x−2≥0

By the rational zeros theorem, the possible rational zeros of the polynomial are LaTeX: \pm1,\:\pm2,\:\pm\frac{1}{2},\:±1,±2,±12,and LaTeX: \pm\frac{1}{4}.±14. Use synthetic division to show that LaTeX: -1\:−1is a zero.

Using synthetic division to show that -1 is a zero of the polynomial.PNG

So now we can partially factor f(x).

LaTeX: f\left(x\right)=\left(x+1\right)\left(4x^3-12x^2+9x-2\right)f(x)=(x+1)(4x3−12x2+9x−2)

Use synthetic division to show that 2 is a zero of the cubic polynomial factor above.

Using synthetic division to show that 2 is a factor of the polynomial.PNG

Now we are able to completely factor f(x).

LaTeX: f\left(x\right)=\left(x+1\right)\left(x-2\right)\left(4x^2-4x+1\right)=\left(x+1\right)\left(x-2\right)\left(2x-1\right)^2f(x)=(x+1)(x−2)(4x2−4x+1)=(x+1)(x−2)(2x−1)2

The zeros of this polynomial are LaTeX: -1,\:\frac{1}{2}\:\left(multiplicity\:2\right),\:−1,12(multiplicity2),and 2. The dominating term is LaTeX: 4x^4.4x4.The y-intercept, found by evaluating f(0), is (0,LaTeX: -2−2).

We can now graph the polynomial function to determine where LaTeX: f\left(x\right)\ge0.f(x)≥0. The degree of f(x) is 4 which is an even number, and the leading coefficient is positive so the end behavior is symbol for end behavior when a is positive and n is even-1.PNGThis means that to the left of the smallest zero and to the right of the largest zero the graph is greater than 0. Then fill in the rest of the graph using the fact that the curve crosses the x-axis at any zero with an odd multiplicity, and bounces off the x-axis for any zero with an even multiplicity.  So, the graph will go from negative touch the x-axis at LaTeX: x=\frac{1}{2}x=12, and then immediately bounce back to more negative values (because of the multiplicity of 2 for the zero at LaTeX: x=\frac{1}{2}x=12) and then the graph will cross the x-axis from negative to the positive at x=2. Thus, the solution set is:

(LaTeX: -\infty,-1−∞,−1]LaTeX: \cup\lbrace\frac{1}{2}\rbrace\cup∪{12}∪[LaTeX: 2,\infty2,∞)

Examples

Here are some videos with examples worked out showing how to solve polynomial inequalities graphically.