# Importance of Coefficients in Polynomials

In any polynomial, the coefficients play a crucial role in determining its characteristics. Let’s see the role of coefficients in determining what kind of graph a linear polynomial will have. Consider the following linear polynomial:

\[a\left( x \right): mx + b\]

The two coefficients in this polynomial are *m* and *b*. How are their values linked to the graph of the polynomial? To understand that, let us first fix the value of *b*, and vary the value of *m*. Let us fix *b* equal to 1, and let us give three different values to *m*: 1, 2 and 5. The following figure shows (on the same graph) the three lines we will obtain:

What do we observe? Increasing the value of *m* increases the *steepness* or *slope* of the line. In fact, when a linear polynomial is expressed as \(mx + b\), the parameter *m* is called the **slope of the line**. Clearly, the value of *m* is strongly related with the graph of the corresponding linear polynomial.

Now, let’s see the role of *b*. Let us now fix the value of *m* at 1, and give three different values to *b*: –2, 0 and 2. The following figure shows (once again on the same graph) the three lines we will obtain:

What do we observe in this case? Increasing the value of *b* *shifts the line upward*, while not having any effect on the steepness of the line. When a linear polynomial is expressed as ** mx + b**, the parameter

*b*is called the

**y-intercept**, since it determines where the line will intersect the

*y*axis. In other words, it determines the

*vertical positioning*of the line.

Clearly, the coefficients in a linear polynomial *mx* + *b* have a strong correlation with its graph. The zero of a linear polynomial ** mx + b** is also decided by the coefficients:

\[x = - \frac{b}{m}\]

Here is another example. The following figure shows the curves corresponding to three different quadratic polynomials. Once again, note how the coefficients determine the position of the curve, its *spread*, and whether it opens upward or downward:

The graph above indicates that if the coefficient of the square term is negative in a quadratic polynomial, the corresponding curve will open downward.

**Example 1: **In the following figure, three lines marked A, B and C are shown:

Which of the three lines has the highest slope, and which has the highest *y* intercept?

**Solution:** The line C seems to be the *steepest*. However, from a lower class, you already know that a line oriented this way will have a *negative slope*. In fact, the equation of line C is:

\[y = - 3x + 2.5\]

Can you deduce this from the graph? Now, among A and B, line A is steeper. Thus, we can say that line A has the highest slope.

The *y* intercept is highest for line C, since its intersection point with the *y* axis is the *topmost*.

**Example 2: **Three linear polynomials are defined as follows:

\[\begin{array}{l}a\left( x \right): & - px + 7\\b\left( x \right): & qx + 3\\c\left( x \right): & rx - 2\end{array}\]

*p*, *q* and *r* are variables. Among the three lines corresponding to these equations, which line will intersect the *y* axis at the lowermost point?

**Solution:** The *y* intercept is lowest for the third polynomial, and hence the third line will intersect the *y* axis at the lowermost point. Note that the values of *p*, *q* and *r* have no effect whatsoever on the *y* intercepts. Those values will only control the steepness of the lines, and not their *vertical positioning*.