Polynomial Inequalities Polynomial Inequalities The first step in solving a polynomial inequality is to find the polynomial's zeroes (its x-intercepts). Between any two consecutive zeroes, the polynomial will be either positive or negative. Since the inequality is asking for positivity ("greater than zero") or negativity ("less than zero"), finding the intercepts ("equal to zero") is the way to get started. If you think of the problem graphically, the zeroes are where the polynomial crosses the x-axis; between any two consecutive crossing-points, the polynomial will either be above the axis (and thus positive) or below it (and thus negative). Polynomial expression can be defined as an expression which contains number of terms and they are joined together by operators. Here we will discuss a Polynomial Inequalities. An expression that consists of constants, variables and exponent values joined together by mathematical operators like addition, subtraction, multiplication, are known as Polynomials.

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Infinite values are not present in case of polynomials. For example: 10xy2 – 5x + 5y3 – 12. Negative Numbers and Fractions can also be included in case of polynomials expressions. Division operator is not used in polynomials. If in any polynomial less than (<) and greater than (>) operator is present then we can say that polynomial has inequality in it. Steps to solve a polynomial are shown below: Step 1: Polynomial has zero on its right hand side. It doesn't matters zero is present either on left side or on right side of a polynomial. Suppose we have a polynomial p2 - 3p – 10 < 0. Step 2: Find the factors of polynomial if possible. Factors of polynomial are: (p - 5) (p + 2) < 0. Step 3: Determine where the value of polynomial is zero or at which Point we get the value of polynomial as zero. We get two values of polynomial i.e. p = - 2 and p = 5. Step 4: Then plot the graph of polynomial where value of polynomial is zero. Find the solution: (x + 4)(x – 2)(x – 7) > 0 Since they've already factored this polynomial, much of my work is already done. So I'll go straight to finding the zeroes: x + 4 = 0, so x = –4 x – 2 = 0, so x = 2 x – 7 = 0, so x = 7

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These three zeroes divide the x-axis into four intervals: (–infinity, –4), (–4, 2), (2, 7), and (7, +infinity). I need to figure out on which of these intervals the polynomial's graph is above the x-axis. If I'd multiplied the factors, I'd have ended up with a positive cubic polynomial, and I know what such a cubic looks like: it starts down on the left, comes up toward the axis, and eventually zooms upward on the right, with a bit of flexure in the middle, something like this:

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