The linear factors of a polynomial are the first-degree equations that are the building blocks of more complex and higher-order polynomials. Linear factors appear in the form of ax + b and cannot be factored further. Each linear factor represents a different line that, when combined with other linear factors, result in different types of functions with increasingly complex graphical representations. The individual elements and properties of a linear factor can help them be better understood.
A linear factor of a polynomial is univariate, meaning it only has one variable that affects the function. Typically, the variable will be designated as x and will correspond to movement on the x-axis. The function will also typically be labeled as y, as in y = ax+b. The values of the variable rely on the real numbers, which are any number to be found on a continuous number line, though for simplicity, the most complex numbers typically used are rational numbers, which are terminating number forms like 2, 0.5 or 1/4.
The slope of a linear factor is the coefficient assigned to the variable in the form y = ax+ b. The a-coefficient predicts the behavior of the inputs in regard to their placement along the x- and y-axes. For example, if the value of a is 5, the value of y will be five times the value of x, meaning that for every forward movement of the x value on the graph, the y value will increase by a factor of 5.
A constant in a linear equation is the b in the form y = ax + b. A linear factor may or may not have a constant in its equation; if there is no constant, it is implied the value of the constant is 0. The constant can move the line either way horizontally on the graph. For example, if the value of b is 2, that means the line will move over two places upwards on the y-axis. This movement is the last computation of the linear factor and on the x variable. When the x value is 0, the constant becomes the y-intercept, where the line crosses the y-axis.
- "An Introduction to Mathematical Thought"; Edward Russell Stabler; 1953
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