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A short class with Sarah and her students. The other ways are under More.
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Welcome class! Today we're diving into polynomials. Can anyone tell me what a polynomial is?
Isn't it an expression with variables like x or y raised to whole-number powers?
Exactly! And the highest power of the variable is called the degree. For example, in 4x + 2, the degree is 1 because the highest power is x^1.
What about other types of polynomials? I heard there are quadratic and cubic ones.
Yes! A polynomial of degree 2 is called a quadratic polynomial, and one of degree 3 is a cubic polynomial. Quadratics take the form ax² + bx + c where a ≠ 0.
Can you give us examples of cubic polynomials?
Sure! An example is 2x³ - 5x² + 3. Any more thoughts?
So all these are polynomials as long as they follow the rules?
Yes, expressions with variables raised to fractional or negative powers aren't polynomials. Now, let's summarize what we've learned: polynomials can be linear, quadratic, or cubic, all defined by their degree.
Next, let’s discuss zeroes of polynomials. Who can tell me what a zero is?
I think it's the value of x that makes the polynomial equal zero.
Correct! For instance, in the polynomial p(x) = x² - 4, the zeroes are the points where p(x) equals zero.
How do we find the zeroes of a polynomial?
Great question! We replace x with various values until we find the points where the polynomial evaluates to zero. For p(x) = x² - 3, we find the zeroes as x equal to the square root of 3.
And this is represented graphically, right?
Absolutely! The graph intersects the x-axis at the zeroes. Let’s sum up: the zeroes are critical to defining the behavior of polynomials and are visible on their graphs.
Now we will relate zeroes to their graphical representation. Why are these zeroes important?
They show where the graph hits the x-axis!
Correct! For a quadratic polynomial, we can have 0, 1 or 2 real zeroes depending on the graph's shape. Can you tell me how the graph looks if there are two zeroes?
It intersects the x-axis at two distinct points.
Exactly! And if it touches at one point?
Then there is one zero, but it appears twice.
Perfect! If the graph does not touch the x-axis, then there are no real zeroes. Always remember that the degree of the polynomial gives you insights into how many zeroes you might find.
Moving on, let’s explore the relationship between zeroes and the coefficients of polynomials. Who remembers the sum and product relationship for quadratic polynomials?
Isn't it that the sum of the zeroes is equal to -b/a?
Exactly! If p(x) = ax² + bx + c, then the zeroes α and β satisfy α + β = -b/a and αβ = c/a.
What about cubic polynomials?
For cubic polynomials, there are similar relationships but involving more terms. The sum of the zeroes of p(x) = ax³ + bx² + cx + d can be expressed as -b/a.
Can you show us an example?
Certainly! For p(x) = 2x³ - 4x² + x - 1, if zeroes are 3, -1, and 1/2, you could calculate the relationships. Remember these formulas help us understand the structure of polynomials!
If x is near, and p(x) is clear, zeroes find their roots, let's give a cheer!
Imagine a tree (x) where branches (coefficients) hold fruits (zeroes). To find the fruits, you must climb the tree, which represents finding the polynomial's roots!
Dodge Zeros Carefully: Degree indicates max Zeros for Each polynomial.
PZC: Polynomials have Zeroes that can be Coefficients.
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What is a polynomial?
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Degree of a polynomial
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Zeroes of a polynomial
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Linear polynomial
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Quadratic polynomial
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An algebraic expression consisting of variables raised to whole-number powers combined with coefficients.
A polynomial of degree 1, which can be expressed in the form ax + b.
A polynomial of degree 2, expressed in the form ax² + bx + c.
A polynomial of degree 3, expressed in the form ax³ + bx² + cx + d.
A value of x for which the polynomial evaluates to zero.
The highest power of the variable in a polynomial.
This section covers the definition of polynomials, their types including linear, quadratic, and cubic polynomials, and the relationship between zeroes and coefficients.
This section introduces polynomials in one variable and their degrees, exemplifying types such as linear, quadratic, and cubic polynomials. It describes how to find zeroes of polynomials and the significance of these zeroes geometrically and algebraically. Additionally, the section discusses the relationship between the coefficients of polynomials and their zeroes.
In this section, we examine polynomials defined as expressions made up of variables raised to whole-number powers, emphasizing their highest degree. The types of polynomials discussed include:
ax + b) and have one root, found where the graph intersects the x-axis.ax² + bx + c), they can have up to 2 roots, indicated by their x-intercepts on a graph.ax³ + bx² + cx + d) can have at most 3 roots.The concept of zeroes is crucial as it indicates values for which the polynomial equals zero. The section illustrates how to compute zeroes using various examples and emphasizes the geometric significance of these zeroes as x-coordinates where the polynomial graph intersects the x-axis. Additionally, it discusses relationships between the zeroes and coefficients of polynomials, such as the sum and product of zeroes for quadratic and cubic polynomials, concluding with the insight that a polynomial of degree n can intersect the x-axis at most at n points.