Question

f(x) = (x + 1)(x-2)(x - 3)²
y
15
-2
4 5
-5
-10
-20
For the single roots -1 and 2, the graph
For the double root 3, the graph
505
10
X
the x-axis at the intercepts.
the x-axis at the intercepts.

Answer 1

The complete statements are:

• For the single roots -1 and 2, the graph crosses the x-axis at the intercepts.
• For the double root 3, the graph touches the x-axis at the intercepts.

Missing part of the question

Complete the blanks for the function f(x) = (x + 1)(x-2)(x - 3)²

How to fill in the blanks?

The function is given as:

f(x) = (x + 1)(x-2)(x - 3)²

Express as products

f(x) = (x + 1) * (x-2) * (x - 3)²

Include the multiplicities

f(x) = (x + 1)¹ * (x-2)¹ * (x - 3)²

The factors that have a multiplicity of 1 are single roots, while the ones with multiplicity of 2 are double roots.

This means that:

• 1 multiplicity = x + 1 and x - 2
• 2 multiplicity = x - 3

The graph crosses the x-axis at 1 multiplicity and it touches the x-axis at 2 multiplicity

Hence, the terms that complete the blanks are crosses and touches, respectively

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