Question

A polynomial function has a root of –4 with multiplicity 4, a root of –1 with multiplicity 3, and a root of 5 with multiplicity 6. If the function has a positive leading co efficient and is of odd degree, which could be the graph of the function?

its B

Answer 1

Therefore, the polynomial function with a leading coefficient of 1 and roots 2i and 3i, each with multiplicity 1, is
`x^4`+13
`x^2`+36.

The polynomial function with a leading coefficient of 1 and roots 2i and 3i, each with multiplicity 1, can be constructed by expressing each root as a factor.

The complex roots 2i and 3i arise from the solutions to the quadratic equation
`x^2`+4=0 and
`x^2`+9=0, respectively.

The factorization of the polynomial is then given by (x−2i)(x+2i)(x−3i)(x+3i). Expanding this expression results in a polynomial function with the desired properties.

Let's simplify the expression step by step:

(x−2i)(x+2i)(x−3i)(x+3i)(x+3i)

= (
`x^2` - (2)
`i^2`)(
`x^2`- (3i)^2)

=(
`x^2`+4)(
`x^2`+9)

=
`x^4`+13
`x^2`+36.

​Question

Which polynomial function has a leading coefficient of 1 and roots 2i and 3i with multiplicity 1?

Answer 2

To draw this graph, we start from the left in quadrant 3 drawing the curve to -4 on the x-axis to touch it but not cross. We continue back down and curve back around to cross the x-axis at -1. We continue up past -1 and curve back down to 5 on the x-axis. We touch here without crossing and draw the rest of our function heading back up. It should form a sideways s shape.

Explanation:

A polynomials is an equation with many terms whose leading term is the highest exponent known as degree. The degree or exponent tells how many roots exist. These roots are the x-intercepts.

This polynomial has roots -4, -1, and 5. This means the graph must touch or cross through the x-axis at these x-values. What determines if it crosses the x-axis or the simple touch it and bounce back? The even or odd multiplicity - how many times the root occurs.

In this polynomial:

Root -4 has even multiplicity of 4 so it only touches and does not cross through.

Root -1 has odd multiplicity of 3 so crosses through.

Root 5 has even multiplicity of 6 so it only touches and does not cross through.

Lastly, what determines the facing of the graph (up or down) is the leading coefficient. If positive, the graph ends point up. If negative, the graph ends point down. All even degree graphs will have this shape.

To draw this graph, we start from the left in quadrant 3 drawing the curve to -4 on the x-axis to touch it but not cross. We continue back down and curve back around to cross the x-axis at -1. We continue up past -1 and curve back down to 5 on the x-axis. We touch here without crossing and draw the rest of our function heading back up. It should form a sideways s shape.