Example: Find the roots of the following polynomial function: a. x(x+7)²(x-1) = 0;Roots are;0, -7 with multiplicity of 2 and 1. b. x³ + 3x²-9x + 5 = 0. Through synthetic division the roots are: (x+5)(x-1)² = 0 ;the roots are: -5, 1, 1 or-5 and 1 with multiplicity of 2. Directions: Find the roots of the following polynomial equation or function. Apply Zero-Product Theorem. 1. x(x+5) = 0 2. (x-1)(x+4)(x-4) = 0 3. x³(x+5)²(x-2)³ = 0 4. x³-13x-12 = 0 5. x((x²-x²-6)See answer

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Example Find the roots of the following polynomial function a x x 7 x 1 0Roots are0 7 with multiplicity of 2 and 1 b x 3x 9x 5 0 Through synthetic division the roots are x 5 x 1 0 the roots are 5 1 1…

Question

Example: Find the roots of the following polynomial function: a. x(x+7)²(x-1) = 0;Roots are;0, -7 with multiplicity of 2 and 1. b. x³ + 3x²-9x + 5 = 0. Through synthetic division the roots are: (x+5)(x-1)² = 0 ;the roots are: -5, 1, 1 or-5 and 1 with multiplicity of 2. Directions: Find the roots of the following polynomial equation or function. Apply Zero-Product Theorem. 1. x(x+5) = 0 2. (x-1)(x+4)(x-4) = 0 3. x³(x+5)²(x-2)³ = 0 4. x³-13x-12 = 0 5. x((x²-x²-6)

Basic Answer

Task Breakdown:

You are asked to find the roots of several polynomial equations using the Zero-Product Theorem. The Zero-Product Theorem states that if the product of several factors equals zero, then at least one of the factors must be zero.

Solution:

Step 1: Identify the factors in each polynomial equation.

• For each polynomial, identify the factors that are multiplied together to form the equation.

Step 2: Set each factor equal to zero.

• For each factor, set it equal to zero and solve for $x$.

Step 3: List the roots.

• Combine the solutions from each factor to list all the roots of the polynomial.

Detailed Solutions:

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

Step 1: Identify the factors

• The factors are $x$ and $(x+5)$.

Step 2: Set each factor equal to zero

• $x=0$
• $x+5=0$ ⟹ $x=-5$

Step 3: List the roots

• The roots are $0$ and $-5$.

Final Answer:

The roots are $0$ and $-5$.

2. $(x-1)(x+4)(x-4)=0$

Step 1: Identify the factors

• The factors are $(x-1)$, $(x+4)$, and $(x-4)$.

Step 2: Set each factor equal to zero

• $x-1=0$ ⟹ $x=1$
• $x+4=0$ ⟹ $x=-4$
• $x-4=0$ ⟹ $x=4$

Step 3: List the roots

• The roots are $1$, $-4$, and $4$.

Final Answer:

The roots are $1$, $-4$, and $4$.

3. $x^3(x+5{)}^2(x-2{)}^3=0$

Step 1: Identify the factors

• The factors are $x^3$, $(x+5{)}^2$, and $(x-2{)}^3$.

Step 2: Set each factor equal to zero

• $x^3=0$ ⟹ $x=0$ (with multiplicity 3)
• $(x+5{)}^2=0$ ⟹ $x=-5$ (with multiplicity 2)
• $(x-2{)}^3=0$ ⟹ $x=2$ (with multiplicity 3)

Step 3: List the roots

• The roots are $0$ (multiplicity 3), $-5$ (multiplicity 2), and $2$ (multiplicity 3).

Final Answer:

The roots are $0$ (multiplicity 3), $-5$ (multiplicity 2), and $2$ (multiplicity 3).

4. $x^3-13x-12=0$

Step 1: Identify the factors

• This polynomial does not factor easily by inspection. We need to use synthetic division or another method to find the roots.

Step 2: Use synthetic division to find roots

• Testing possible rational roots (factors of the constant term over factors of the leading coefficient):
  • Possible roots: $\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12$
  • Testing $x=-1$:
    $$\begin{array}{rrrrr}-1& 1& 0& -13& -12\\ & & -1& 1& -12\\ & 1& -1& 12& 0\end{array}$$
  • The remainder is zero, so $x=-1$ is a root.
  • The quotient is $x^2-x-12$.

Step 3: Factor the quotient

• $x^2-x-12=(x-4)(x+3)$

Step 4: List the roots

• The roots are $-1$, $4$, and $-3$.

Final Answer:

The roots are $-1$, $4$, and $-3$.

5. $x((x^2-x^2-6))$

Step 1: Simplify the expression

• $x((x^2-x^2-6))=x(0-6)=-6x$

Step 2: Set the factor equal to zero

• $-6x=0$ ⟹ $x=0$

Step 3: List the roots

• The root is $0$.

Final Answer:

The root is $0$.