**Zeros in Quadratics and Other Polynomials**

One of the most crucial skills in solving quadratic problems is understanding the language used. While in real life these are slightly different terms, when referring to a single quadratic, these words all mean the same thing:

Zeros = Roots = Solutions = Real Solutions = *x*-intercepts

So when a question asks you about the zeros of a quadratic, they are asking about the *x*-intercepts.

The first graph has no intercepts, and thus no real solutions. The second is tangent to the *x*-axis, so it has one solution at *x *= 2. The third graph has solutions at *x* = 1 and *x* = 3.

**Factors**

The first graph above shows an equation that would not factor.

The second graph shows an equation where (*x* – 2) MUST be a factor, because *y* = 0 when *x *= 2.

The third graph shows an equation where (*x *-1) and (*x* – 3) MUST be factors.

If given a problem like:

An equation has roots at -1 and 5. Which if the following must be a factor?

A) (x+5)

B) (x)

C) (x-1)

D) (x+1)

**Strategy****:**

**Rebuild the factored equation by forming binomials with the given roots (solutions).**

If an equation has roots of -1 and 5, then in its factored form it would look like ((x+1)(x-5)). This is because those binomials have solutions of -1 and 5. Thus, the answer would be D.

**Zeros in Other Equations**

The exact same concept applies if you are asked about zeros in other equations. For instance, given the graph below:

You can easily see that it has 4 x-axis intercepts, and thus would have 4 zeros or solutions.

Note that the approach above for identifying factors holds true regardless of the number of roots / solutions. If given a polynomial like:

(ax^3+bx^2+cx+d=0) with roots of -4, -2, and 3,

you would apply exactly the same logic.

If you know two zeros of a quadratic, then you actually also know the

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