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## Algebra 2

### Course: Algebra 2>Unit 5

Lesson 2: Positive and negative intervals of polynomials

# Multiplicity of zeros of polynomials

The polynomial p(x)=(x-1)(x-3)² is a 3rd degree polynomial, but it has only 2 distinct zeros. This is because the zero x=3, which is related to the factor (x-3)², repeats twice. This is called multiplicity. It means that x=3 is a zero of multiplicity 2, and x=1 is a zero of multiplicity 1. Multiplicity is a fascinating concept, and it is directly related to graphical behavior of the polynomial around the zero.

## Want to join the conversation?

• Quick question out of curiosity, how would you be able to find out how the graph of a polynomial comes from the third quadrant (bottom left) instead of the second (top left) as the x axis approaches negative infinity? • Remember that the graph of a function can 'bend' n-1 times, where n is the degree of the polynomial. After plotting the 'bends', plug in large negative values for x. If the y values are trending towards negative infinity as well, the function will come from the third quadrant. If the y values are increasing, it will come from the second quadrant. Or, if you know the end behavior on the positive end, you could determine whether it is an even or an odd function.
• I am still confused why an even multiplicity will result in no sign change, while an odd multiplicity will. Could someone explain? • You can break a polynomial into "linear factors." For example, we can break x^3 - 4x into (x + 2)(x)(x - 2).

Imagine you are driving along the number line from left to right. As you drive onto the screen from the left, all three factors will be negative numbers.

For example, if x = -100, the polynomial will equal (-100 + 2)(-100)(-100 - 2) = (negative)(negative)(negative). When you multiply three negative numbers together, you get a negative result, so the entire polynomial will come out negative.

Now imagine you cross x = -2. The first linear factor, (x + 2), goes from negative to zero to positive. The very instant you cross x = 2, the polynomial becomes (positive)(negative)(negative) = positive.

Every time you "drive across" a zero, exactly one of the linear factors changes sign from negative to positive, and that flips the sign of the polynomial.

But when you have two identical roots, then TWO of the factors change sign from negative to positive at the same instant. So in that case, the sign of the polynomial DOESN'T change.

For example, say we have x^3 + 2x^2 = (x + 2)(x)(x). When we drive in from the left, the three factors start out as (negative)(negative)(negative), so the polynomial is negative. When we drive across x = -2, the factors become (positive)(negative)(negative), so the polynomial becomes positive.

But when we drive across x = 0, BOTH of the remaining factors flip at the same instant, so the factors become (positive)(positive)(positive), and the polynomial stays positive.
• what is multiplicity • why is there a grey dot at x=1.65 on the white graph • Did he spell zeroes wrong i dont know if its an american thing or not bold • If p(x)= (x-1) (7x-21) (x-3)
here zeros are 2 since 2 parts of the factored expression point to the same zero. But, (7x-21) & (x-3) are different expressions as the latter is a scaled-down version of the first one. Will it make any difference to the idea of multiplicity? • Is the reason that a multiplicity of 2 curves upwards because any number that is exponentiated to the power of an even number cannot be negative? EX: x^2 can never be a neg #. • I got most of the video, but the end just completely flew me off track. What does Sal mean by the number of zeros is equal to degree of a polynomial? Please help.
(1 vote) • The degree of a polynomial is the highest exponent that appears in it. The degree of x³-5x²+1 is 3.

A zero of a polynomial is a value that you can plug in for x to make the whole expression equal 0. -1 is a zero of the polynomial x⁵+1, since (-1)⁵+1=0.

Most polynomials have multiple different zeroes. 1 and 2 are both zeroes of x²-3x+2. Sal is saying that any polynomial has exactly as many zeroes as its degree, assuming you count the multiplicity of zeroes and allow complex zeroes.  