7 THE ROOTS, OR ZEROS,

P(1)  =  5· 1^{3} − 4· 1^{2} + 7· 1 − 8 
=  5 − 4 + 7 − 8  
=  0. 
It is traditional to speak of a root of a polynomial. Of a function in general, we speak of a zero.
Example 2. The roots of this quadratic
x^{2} −x − 6 = (x + 2)(x − 3)
are −2 and 3. Those are the values of x that will make the polynomial equal to 0.
3. What are the xintercept and yintercept of a graph?
The xintercept is that value of x where the graph crosses or touches the xaxis. At the xintercept  on the xaxis  y = 0.
The yintercept is that value of y where the graph crosses the yaxis. At the yintercept, x = 0.
4. What is the relationship between the root of a polynomial
4. and the xintercepts of its graph?
The roots are the xintercepts!
The roots of x^{2} −x − 6 are −2 and 3. Therefore, the graph of
y = x^{2} −x − 6
will have the value 0  it will cross the xaxis  at −2 and 3.
5. How do we find the xintercepts of the graph of any function
5. y = f(x)?
Solve the equation, f(x) = 0.
Problem 1.
a) Find the root of the polynomial 2x + 10.
We must solve the equation,
2x + 10 = 0. The solution is x = −5.
b) Where is the xintercept of the graph of y = 2x + 10?
At x = −5. The xintercept is the root.
Problem 2.
a) If a product of factors is 0  if ab = 0  then what may we conclude
a) about the factors a, b ?
Either a = 0, or b = 0.
b) Name the roots of this polynomial:
f(x) = (x + 4)(x + 2)(x − 1)
−4, −2, 1.
c) Sketch the graph of f(x). That is, sketch a continuous curve and show
c) its xintercepts.
The xintercepts are the roots.
As for the yintercept, it is the value of y when x = 0. Therefore, the yintercept of a polynomial is simply the constant term, which is the product of the constant terms of all the factors. (See Topic 6, Example 9.)
As for finding the turning points, that hill and valley, that will have to wait for calculus.
For roots of polynomials of degree greater than 2, see Topic 13.
Next Topic: The slope of a straight line
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