Fresh features from the #1 AI-enhanced learning platform.Try it free
Fresh features from the #1 AI-enhanced learning platformCrush your year with the magic of personalized studying.Try it free
Question

# Write a polynomial equation of least degree for each set of roots. Does the equation have an odd or even degree? How many times does the graph of the related function cross the x-axis? -2, -0.5, 4

Solution

Verified
Step 1
1 of 2

The corollary to the fundamental theorem of algebra states:

Every polynomial $P(x)$ with the degree $n$ ($n>0$) can be written as the product of a constant $k\neq 0$ and $n$ linear factors :

$P(x)=k(x-r_1)(x-r_2)\cdots(x-r_{n-1})(x-r_n)$

Thus a polynomial equation of degree $n$, has $n$ complex roots, namely $r_1,r_2,\cdots,r_n$.

Having $-2$,$-0.5$ and 4 as roots of a polynomial with the least degree means:

$P(x)=(x+2)(x-0.5)(x-4)$

The polynomial crosses the x-axis as many times as there are real roots, therefore 3 times.

## Recommended textbook solutions #### Precalculus

2nd EditionISBN: 9780076602186 (1 more)Carter, Cuevas, Day, Malloy
8,886 solutions #### Advanced Mathematical Concepts: Precalculus with Applications

1st EditionISBN: 9780078682278Carter, Cuevas, Holliday, Marks
7,568 solutions #### Precalculus with Limits

3rd EditionISBN: 9781133962885 (2 more)Larson
11,142 solutions #### Precalculus: Mathematics for Calculus

7th EditionISBN: 9781305071759 (3 more)Lothar Redlin, Stewart, Watson
9,606 solutions