# Lesson 14

What Do You Know About Polynomials?

- Let's put together what we've learned about polynomials so far.

### Problem 1

We know these things about a polynomial function, \(f(x)\): it has exactly one relative maximum and one relative minimum, it has exactly three zeros, and it has a known factor of \((x-4)\). Sketch a graph of \(f(x)\) given this information.

### Problem 2

Mai graphs a polynomial function, \(f(x)\), that has three linear factors \((x+6)\), \((x+2)\), and \((x-1)\). But she makes a mistake. What is her mistake?

### Problem 3

Here is the graph of a polynomial function with degree 4.

Select **all** of the statements that are true about the function.

The leading coefficient is positive.

The constant term is negative.

It has 2 relative maximums.

It has 4 linear factors.

One of the factors is \((x-1)\).

One of the zeros is \(x=2\).

There is a relative minimum between \(x=1\) and \(x=3\).

### Problem 4

State the degree and end behavior of \(f(x)=2x^3-3x^5-x^2+1\). Explain or show your reasoning.

### Problem 5

Is this the graph of \(g(x)=(x-1)^2(x+2)\) or \(h(x)=(x-1)(x+2)^2\)? Explain how you know.

### Problem 6

Kiran thinks he knows one of the linear factors of \(P(x)=x^3 + x^2 - 17 x + 15 \). After finding that \(P(3)=0\), Kiran suspects that \(x-3\) is a factor of \(P(x)\), so he sets up a diagram to check. Here is the diagram he made to check his reasoning, but he set it up incorrectly. What went wrong?

\(x^2\) | \(4x\) | -5 | |
---|---|---|---|

\(x\) | \(x^3\) | \(4x^2\) | \(\text-5x\) |

3 | \(3x^2\) | \(12x\) | 15 |

### Problem 7

The polynomial function \(B(x)=x^3+8x^2+5x-14\) has a known factor of \((x+2)\). Rewrite \(B(x)\) as a product of linear factors.