How should I think about "multiplicity" for polynomial roots?

Question

I'm having trouble conceptualizing the idea of multiplicity when it comes to finding roots of polynomials. Consider the example from Multiplicity on Wikipedia:

The polynomial $p(x) = x^3 + 2x^2 - 7x + 4$ has $1$ and $-4$ as roots, and can be written as $p(x) = (x+4)(x - 1)^2$.

I understand that, in an algebraic sense, there are three monomials here, each representing a root:

• $(x+4) \Rightarrow -4$
• $(x-1) \Rightarrow 1$
• $(x-1) \Rightarrow 1$

What I don't understand is that, since the graph of this polynomial intersects the x axis only twice, why do we say there are three solutions to $x^3 + 2x^2 - 7x + 4 = 0$?

Answer 1

To reduce the polynomial to a quotient of 1 with a remainder of 0, it is necessary to divide by x-1 twice, hence the multiplicity of 2 for that root. You were on the correct route.

expr = x^3 + 2*x^2 - 7*x + 4

4 - 7*x + 2*x^2 + x^3

PolynomialRemainder[expr, x + 4, x]

0

expr = PolynomialQuotient[expr, x + 4, x]

1 - 2*x + x^2

PolynomialRemainder[expr, x - 1, x]

0

expr = PolynomialQuotient[expr, x - 1, x]

-1 + x

PolynomialRemainder[expr, x - 1, x]

0

expr = PolynomialQuotient[expr, x - 1, x]

1