Question

Learning Task 1. Complete the table below by identifying the degree and real roots of polynomial equations (if a root occurs twice then use multiplicity 2 or if it occurs thrice use multiplicity 3) Three problems are done for you as ex-ample.

Polynomial Equations

4. (x-2) (x + 2) (x - 4 = 0
5. x(x - 1)^2 (x + 2) = 0
6. (x +4)^2 (x - 1)^3 = 0
7. (x + 5) (x + 1) = 0
8. x^2(x - 1) (x + 1) = 0

DEGREE:
ROOTS:
NUMBER OF REAL ROOTS:​

Answer 1

Answer:

4)3 , 5) 4, 6) 6, 7) 2 , 8) 4

Answer 2

Answer:

The answer is given below as image attached as a file

Step-by-step explanation:

A Polynomial Equation is of the form

$a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\ldots+a_{0}=0$ where $n$is the degree of the polynomial equation.

Definition of Terms

• Degree of the Equation- This is the highest exponent that is present in the given function.
• Roots of the Equation- This is the value of $x$ when the function is zero. Sometimes this is called the zeros of the equation.
• Multiplicity. This is the number of times that a factor appears in the factored form of the equation.

4. $(x-2)(x+2)(x-4)=0$

The degree of the equation can be solved by multiplying all the variables from the equation.

$x \cdot x \cdot x=x^{3}$

The degree of the equation should be 3 .

The roots of the equation cabn be solved by applying the zero product property or setting each factor to zero and solve for the unknown variable.

$x-2=0 \\ x+2=0 \\ x-4=0 \\ x=2 \\ x=-2 \\ x=4$

Therefore the roots are$2,-2$ and 4 . Simply count the number of roots solved. The number of roots is 3 .

5. $x(x-1)^{2}(x+2)=0$

For the degree, and the number of roots of the equation, follow the steps in the solution of number 4 .

Similar to the solutions above, set each factor to zero and solve for the unknown variable.

$\begin{array}{rlrl}x+2=0 \\x=0 &\\ x =-2\end{array}$

If the equation is in the form of $(x+a)^{n}(x-a)$, then it said to be that the one of the roots is$-a$ multiplicity of $n$. Thus, one of the zeros of the given function in item number 5 is 1 multiplicity $2 .$

6. $(x+4)^{2}(x-1)^{3}=0$

Degree: 5

Roots: -4 multiplicity 2,1 multiplicity 3

Number of Roots: 5

7. $(x+5)(x+1)=0$

Degree: 2

Roots: $-5 and -1$

Number of Roots: 2

8. $x^{2}(x-1)(x+1)=0$

Degree: 4

Roots: 0 multiplicity 2,1 , and $-1$

Number of Roots: 4