Answer :
Sure! Let's discuss the questions step-by-step:
### Characteristics of [tex]\(g(x) = (x-3)(x+2)(x+3)\)[/tex] and Expansion Strategy
1. Characteristics and Expansion:
The given polynomial [tex]\(g(x) = (x-3)(x+2)(x+3)\)[/tex] is a cubic polynomial because it is the product of three linear factors. This means the highest power of [tex]\(x\)[/tex] will be [tex]\(x^3\)[/tex]. To expand this polynomial, we can multiply the factors step by step.
[tex]\[ (x-3)(x+2)(x+3) \][/tex]
Let's first expand the product [tex]\((x+2)(x+3)\)[/tex]:
[tex]\[ (x+2)(x+3) = x^2 + 5x + 6 \][/tex]
Now, we multiply this result by the remaining factor [tex]\((x-3)\)[/tex]:
[tex]\[ (x-3)(x^2 + 5x + 6) = x(x^2 + 5x + 6) - 3(x^2 + 5x + 6) \][/tex]
Distributing [tex]\(x\)[/tex] and [tex]\(-3\)[/tex] respectively:
[tex]\[ = x^3 + 5x^2 + 6x - 3x^2 - 15x - 18 \][/tex]
Combining like terms, we get:
[tex]\[ = x^3 + 2x^2 - 9x - 18 \][/tex]
So, the expanded form of [tex]\(g(x)\)[/tex] is [tex]\(x^3 + 2x^2 - 9x - 18\)[/tex].
Key Features:
- The highest degree term is [tex]\(x^3\)[/tex], which confirms it is a cubic polynomial.
- The polynomial has three roots: [tex]\(x = 3\)[/tex], [tex]\(x = -2\)[/tex], and [tex]\(x = -3\)[/tex].
### Intercepts of the Polynomial [tex]\(x^3 - 4x^2 = 0\)[/tex]:
2. Intercepts of [tex]\(x^3 - 4x^2 = 0\)[/tex]:
To find the intercepts, we set the equation to zero and solve for [tex]\(x\)[/tex]:
[tex]\[ x^3 - 4x^2 = 0 \implies x^2(x - 4) = 0 \][/tex]
Using the Zero Product Property:
[tex]\[ x^2 = 0 \implies x = 0 \quad \text{(double root)} \][/tex]
[tex]\[ x - 4 = 0 \implies x = 4 \][/tex]
Hence, the roots are [tex]\(x = 0\)[/tex] (with multiplicity 2) and [tex]\(x = 4\)[/tex]. Although a cubic polynomial can have up to three x-intercepts, in this case, [tex]\(x = 0\)[/tex] only counts as one intercept on the graph, even though its multiplicity is 2.
### Can a Cubic Equation Have Four X-Intercepts?
3. Four X-Intercepts in a Cubic Equation:
No, a cubic equation cannot have four x-intercepts. The maximum number of x-intercepts a cubic polynomial can have is three because it is a polynomial of degree 3 and the Fundamental Theorem of Algebra states that a polynomial of degree [tex]\(n\)[/tex] can have at most [tex]\(n\)[/tex] roots (including multiplicities).
### Zero Product Property and Polynomial Roots
4. Zero Product Property:
The Zero Product Property states that if a product of factors equals zero, at least one of the factors must be zero. This property supports the determination of roots in polynomials. When we factored [tex]\(x^3 - 4x^2\)[/tex] into [tex]\(x^2(x-4) = 0\)[/tex], applying this property directly allowed us to find the roots ([tex]\(x = 0\)[/tex] and [tex]\(x = 4\)[/tex]).
### Degree, Roots, and Factors of a Polynomial
5. Relationship Between Degree, Roots, and Factors:
- The degree of a polynomial is the highest power of the variable in the polynomial.
- The number of roots (including multiplicity) of a polynomial is equal to its degree. Thus, a cubic polynomial (degree 3) has exactly 3 roots.
- A fully factored polynomial of degree [tex]\(n\)[/tex] will have [tex]\(n\)[/tex] linear factors corresponding to its [tex]\(n\)[/tex] roots. For example, [tex]\(x^3 - 4x^2\)[/tex] factors to [tex]\(x^2(x - 4)\)[/tex], which has 3 total linear factors when considering multiplicity (two factors of [tex]\(x\)[/tex] and one factor of [tex]\(x-4\)[/tex]).
In summary, the given cubic polynomial exemplifies the core properties of polynomial equations, including the method for expansion, the interpretation of roots and intercepts, and the fundamental principles that govern polynomial behavior.
### Characteristics of [tex]\(g(x) = (x-3)(x+2)(x+3)\)[/tex] and Expansion Strategy
1. Characteristics and Expansion:
The given polynomial [tex]\(g(x) = (x-3)(x+2)(x+3)\)[/tex] is a cubic polynomial because it is the product of three linear factors. This means the highest power of [tex]\(x\)[/tex] will be [tex]\(x^3\)[/tex]. To expand this polynomial, we can multiply the factors step by step.
[tex]\[ (x-3)(x+2)(x+3) \][/tex]
Let's first expand the product [tex]\((x+2)(x+3)\)[/tex]:
[tex]\[ (x+2)(x+3) = x^2 + 5x + 6 \][/tex]
Now, we multiply this result by the remaining factor [tex]\((x-3)\)[/tex]:
[tex]\[ (x-3)(x^2 + 5x + 6) = x(x^2 + 5x + 6) - 3(x^2 + 5x + 6) \][/tex]
Distributing [tex]\(x\)[/tex] and [tex]\(-3\)[/tex] respectively:
[tex]\[ = x^3 + 5x^2 + 6x - 3x^2 - 15x - 18 \][/tex]
Combining like terms, we get:
[tex]\[ = x^3 + 2x^2 - 9x - 18 \][/tex]
So, the expanded form of [tex]\(g(x)\)[/tex] is [tex]\(x^3 + 2x^2 - 9x - 18\)[/tex].
Key Features:
- The highest degree term is [tex]\(x^3\)[/tex], which confirms it is a cubic polynomial.
- The polynomial has three roots: [tex]\(x = 3\)[/tex], [tex]\(x = -2\)[/tex], and [tex]\(x = -3\)[/tex].
### Intercepts of the Polynomial [tex]\(x^3 - 4x^2 = 0\)[/tex]:
2. Intercepts of [tex]\(x^3 - 4x^2 = 0\)[/tex]:
To find the intercepts, we set the equation to zero and solve for [tex]\(x\)[/tex]:
[tex]\[ x^3 - 4x^2 = 0 \implies x^2(x - 4) = 0 \][/tex]
Using the Zero Product Property:
[tex]\[ x^2 = 0 \implies x = 0 \quad \text{(double root)} \][/tex]
[tex]\[ x - 4 = 0 \implies x = 4 \][/tex]
Hence, the roots are [tex]\(x = 0\)[/tex] (with multiplicity 2) and [tex]\(x = 4\)[/tex]. Although a cubic polynomial can have up to three x-intercepts, in this case, [tex]\(x = 0\)[/tex] only counts as one intercept on the graph, even though its multiplicity is 2.
### Can a Cubic Equation Have Four X-Intercepts?
3. Four X-Intercepts in a Cubic Equation:
No, a cubic equation cannot have four x-intercepts. The maximum number of x-intercepts a cubic polynomial can have is three because it is a polynomial of degree 3 and the Fundamental Theorem of Algebra states that a polynomial of degree [tex]\(n\)[/tex] can have at most [tex]\(n\)[/tex] roots (including multiplicities).
### Zero Product Property and Polynomial Roots
4. Zero Product Property:
The Zero Product Property states that if a product of factors equals zero, at least one of the factors must be zero. This property supports the determination of roots in polynomials. When we factored [tex]\(x^3 - 4x^2\)[/tex] into [tex]\(x^2(x-4) = 0\)[/tex], applying this property directly allowed us to find the roots ([tex]\(x = 0\)[/tex] and [tex]\(x = 4\)[/tex]).
### Degree, Roots, and Factors of a Polynomial
5. Relationship Between Degree, Roots, and Factors:
- The degree of a polynomial is the highest power of the variable in the polynomial.
- The number of roots (including multiplicity) of a polynomial is equal to its degree. Thus, a cubic polynomial (degree 3) has exactly 3 roots.
- A fully factored polynomial of degree [tex]\(n\)[/tex] will have [tex]\(n\)[/tex] linear factors corresponding to its [tex]\(n\)[/tex] roots. For example, [tex]\(x^3 - 4x^2\)[/tex] factors to [tex]\(x^2(x - 4)\)[/tex], which has 3 total linear factors when considering multiplicity (two factors of [tex]\(x\)[/tex] and one factor of [tex]\(x-4\)[/tex]).
In summary, the given cubic polynomial exemplifies the core properties of polynomial equations, including the method for expansion, the interpretation of roots and intercepts, and the fundamental principles that govern polynomial behavior.
