Learning objectives
The zeros of a function are the input values which make the function equal to zero. This means a is a zero of f\left(x\right) if f\left(a\right) = 0. We also refer to these solutions as roots of the equation f\left(x\right)=0. These zeros or roots appear on the graph of the function as x-intercepts.
Real zeros can also serve as endpoints for intervals that satisfy polynomial inequalities, as they represent points where the polynomial function changes its sign. This concept helps in finding intervals where the polynomial function is greater than or less than a certain value.
The fundamental theorem of algebra says the number of complex roots of any polynomial is equal to the degree of the polynomial. Remember that complex roots refer to real and imaginary roots. The real zeros of a function will be the x-intercepts of its graph.
Recall the factor theorem says if x=a is a root of f\left(x\right) = 0, then \left(x-a\right) is a factor of f\left(x\right).
The multiplicity of a zero is the number of times that its corresponding factor appears in the function. The multiplicities of the zeros in the function will sum to the degree of the polynomial by the fundamental theorem of algebra. Zeros with different multiplicities look different graphically.
A root of multiplicity 1 crosses through the x-axis with no point of inflection. A root with an odd multiplicity greater than 1 crosses through the x-axis with a point of inflection. Roots with even multiplicity are tangent to the axis which means they touch the x-axis, then change direction and do not cross the x-axis.
When the coefficients of a polynomial meet certain criteria, complex roots and irrational roots will come in conjugate pairs.
A polynomial f\left(x\right)=x^5+7x^4+17x^3+47x^2+72x-144 has zeros at x=-4 and x=-3i.
Determine the remaining zeros of f\left(x\right).
State the multiplicities of the zeros.
Consider the polynomial function: h\left(x\right)=\left(x-3\right)\left(x^2-4x+5\right).
State the zeros of h\left(x\right).
Write an equation for h\left(x\right) using linear factors.
Given the polynomial function: p\left(x\right)=\left(x+3\right)^2\left(x-4\right)\left(x+5\right), without graphing, describe the behavior of the graph near the zero x=3.
The multiplicity of a zero is the number of times that its corresponding factor appears in the function.
Even multiplicity: function touches the x-axis at that zero then changes direction
Odd multiplicity: function crosses the x-axis at that zero
Complex roots come in conjugate pairs of the form a+bi and a-bi.
The degree of a polynomial function is the highest power of the independent variable in its expression. To determine the degree of a polynomial function, we can examine the successive differences of output values over equal-interval input values. The degree is equal to the least value n for which the successive nth differences are constant.
Consider the function represented by the following table:
A polynomial function, w\left(x\right), is shown in the table:
x | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|
w(x) | 1 | 6 | 31 | 100 | 229 | 426 |
Find successive differences by subtracting consecutive y-values. The degree of a polynomial is found by identifying which successive difference is constant.