**High School Number Sense Lesson 41: Multiplying the Roots of a Polynomial**

Working with polynomials is a common theme on number sense tests. A couple of weeks ago we looked at finding the roots (solutions) of a polynomial with

**SolveX**and

**SolveXExpo**. Today we will look at finding the product of the roots (without having to find the roots themselves). This concept appeared

**9 times**this year, with a median placement at

**question # 42**.

**Number Dojo Level: 302**

To master this concept, we first have to understand what is meant by the

**degree**of the polynomial. This refers to the

**highest power**of the variable. For example:

**x**- 2x + 5 = 0

^{2}**2**, because the highest power of x is 2.

**x**+ 7x

^{3}^{2}- 5x + 12 = 0

**3**, because the highest power of x is 3.

**x**+ x

^{4}^{3}- 4x

^{2}+ x - 5 = 0

**4**, because the highest power of x is 4, and so on.

Degree of 2:

Degree of 2:

- For a polynomial (of degree 2) written as:

**Ax**,

^{2}+ Bx + C = 0**C/A**.

**Degree of 3:**

- For a polynomial (of degree 3) written as:

**Ax**,

^{3}+ Bx^{2}+ Cx + D = 0**-D/A**.

**Degree of 4:**

- For a polynomial (of degree 4) written as:

**Ax**,

^{4}+ Bx^{3}+ Cx^{2}+ Dx + E = 0**E/A**.

**In general:**

- For any polynomial with an
**even**-powered degree (2, 4, etc.), the product of the roots is equal to the**constant****divided by the highest-powered coefficient**. - For any polynomial with an
**odd**-powered degree (1, 3, etc.), the product of the roots is equal to the**negative constant****divided by the highest-powered coefficient**.

**Example 1:**

**The product of the roots of 4x**

^{2}- 3x + 8 = 0 is ___1. Written as Ax

^{2}+ Bx + C = 0, A =

**4**and C =

**8**.

2. The product of the roots is

**C/A**, which is

**8/4**=

**2**.

**Example 2:**

**The product of the roots of 7x**

^{2}+ 5x - 3 = 0 is ___1. Written as Ax

^{2}+ Bx + C = 0, A =

**7**and C =

**-3**.

2. The product of the roots is

**C/A**, which is

**-3/7**.

**Example 3:**

**The product of the roots of 4x**

^{2}+ 5x = 2 is ___1. Written as Ax

^{2}+ Bx + C = 0, this is

**4x**. A =

^{2}+ 5x - 2 = 0**4**and C =

**-2**.

2. The product of the roots is

**C/A**, which is

**-2/4**=

**-1/2**.

**Example 4:**

**The product of the roots of 5x**

^{3}- 7x^{2}+ 2x + 10 = 0 is ___1. Written as Ax

^{3}+ Bx

^{2}+ Cx + D = 0, A =

**5**and D =

**10**.

2. The product of the roots is

**-D/A**, which is

**-10/5**=

**-2**.

**Example 5:**

**The product of the roots of -8x**

^{3}+ 3x^{2}+ 4x - 2 = 0 is ___1. Written as Ax

^{3}+ Bx

^{2}+ Cx + D = 0, A =

**-8**and D =

**-2**.

2. The product of the roots is

**-D/A**, which is

**-(-2)/(-8)**=

**2/(-8)**=

**-1/4**.

**Example 6:**

**The product of the roots of 3x**

^{4}- x^{3}+ 4x^{2}+ 10x - 5 = 0 is ___1. Written as Ax

^{4}+ Bx

^{3}+ Cx

^{2}+ Dx + E = 0, A =

**3**and E =

**-5**.

2. The product of the roots is

**E/A**, which is

**-5/3**.

**Here's a free worksheet to help you practice MultRoots:**

multroots.pdf |

**Up Next for High School: AddRoots**