3 - Polynomials in Factored Form
MHF4U – Polynomials
Date: _________________________
Polynomials in Factored Form
To find zeros/real roots/x-intercepts of a polynomial function of degree 2 or above, the
function has to be fully factored first.
Example 1: Find the zero of f (x ) = 3x − 8
Example 2: Find the zeros of the following function f (x ) = x 2 − 4x − 12
The maximum number of real zeroes = degree = n
At least 0 for even degree functions and at least 1 for odd degree functions
Example 3: Given the function y = x(x − 3 )(x + 2)(x + 1)
a) Determine the:
degree of the polynomial
_______
sign of the leading coefficient
_______
end behaviour:
_________________________________________
y-intercept
_______
x intercepts
________________
zeros
________________
When a polynomial function is in factored form, the zeros can be easily determined from
the factors.
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3 - Polynomials in Factored Form
MHF4U – Polynomials
b) Sketch.
The Nature of Roots
1st order roots: the function crosses the x-axis just like a straight line.
Example: y = x(x − 1)(x + 2)
2nd order roots: a result of a squared factor, the function touches the x-axis like a
parabola.
2
Example: y = (x + 1)(x − 2)
3rd order roots: a result of a cubed factor, the function crosses the x-axis like a cubic.
3
Example: y = (x + 1)(x − 2)
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3 - Polynomials in Factored Form
MHF4U – Polynomials
Example 4: Describe the shape of the graph near the zeros:
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3 - Polynomials in Factored Form
MHF4U – Polynomials
2
Example 5: Given y = −2(x − 1) (x + 2)
a) Find the:
degree of the polynomial
__________
sign of leading coefficient __________
end behaviour
_____________________________________________
y-intercept
__________
x-intercepts
__________
b) Sketch.
Homework
Page 146
1, 2, 6acde, 10abc
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