- The degree of \(f(x)\) is the sum of the exponents: \(7 + 1 + 9 + 9 = 26\).
- The leading coefficient is the constant factor 12.
- The x-intercepts are \((1, 0), (4, 0), (8, 0), (-8, 0)\).
- The y-intercept is \((0, -864691128455135232)\).
- The multiplicities of the zeros \(1, 4, 8, -8\) are \(7, 1, 9, 9\) respectively.
- Final Answers:
- Degree: \(\boxed{26}\)
- Leading Coefficient: \(\boxed{12}\)
- X-intercepts: \(\boxed{(1, 0), (4, 0), (8, 0), (-8, 0)}\)
- Y-intercept: \(\boxed{(0, -864691128455135232)}\)
- Multiplicities: (a) \(\boxed{7}\), (b) \(\boxed{1}\), (c) \(\boxed{9}\), (d) \(\boxed{9}\)
Explanation
- Problem Analysis
The given polynomial function is \(f(x) = 12(x-1)^7(x-4)(x-8)^9(x+8)^9\). This function is expressed as a product of a constant and several linear factors, each raised to a specific positive integer power (the multiplicity). To analyze its properties, we consider its degree, leading coefficient, roots, and y-intercept.
- Degree of the Function
The degree of a polynomial, when written as a product of linear factors, is the sum of the exponents of all these factors. The exponents in the function are 7, 1, 9, and 9. Therefore, the degree of \(f(x)\) is \(7 + 1 + 9 + 9 = 26\).
- Leading Coefficient
The leading coefficient is the coefficient of the highest-degree term. Since each factor \((x-r)\) has a coefficient of 1 for the \(x\) term, the leading coefficient is simply the constant multiplier in front of the expression, multiplied by the coefficient of \(x\) in each factor raised to its respective power. Here, it is \(12 \times 1^7 \times 1^1 \times 1^9 \times 1^9 = 12\).
- X-intercepts
The x-intercepts occur where the function equals zero. Setting \(f(x) = 12(x-1)^7(x-4)(x-8)^9(x+8)^9 = 0\), we find the roots to be \(x = 1, x = 4, x = 8,\) and \(x = -8\). In ordered pair form, these are \((1, 0), (4, 0), (8, 0),\) and \((-8, 0)\).
- Y-intercept
The y-intercept occurs at the point \((0, f(0))\). Evaluating the function at \(x=0\):
\[f(0) = 12(0-1)^7(0-4)(0-8)^9(0+8)^9\]
\[f(0) = 12(-1)^7(-4)(-8)^9(8)^9\]
\[f(0) = 12(-1)(-4)(-1 \times 8)^9(8)^9\]
\[f(0) = 48 \times (-1)^9 \times 8^9 \times 8^9\]
\[f(0) = 48 \times (-1) \times 8^{18} = -48 \times 8^{18} = -864691128455135232\]
Thus, the y-intercept is \((0, -864691128455135232)\).
- Multiplicity of Zeros
The multiplicity of a zero is the exponent of its corresponding factor in the factored form:
AFor the zero \(x = 1\), the factor is \((x-1)^7\), so the multiplicity is 7.
BFor the zero \(x = 4\), the factor is \((x-4)^1\), so the multiplicity is 1.
CFor the zero \(x = 8\), the factor is \((x-8)^9\), so the multiplicity is 9.
DFor the zero \(x = -8\), the factor is \((x+8)^9\), so the multiplicity is 9.
Examples
Polynomials are fundamental in modeling real-world phenomena. For instance, in engineering, they can represent the shape of a bridge arch or the trajectory of an object in flight. By determining the zeros and their multiplicities, an engineer can understand the points where the model crosses a specific baseline, helping to optimize structures or predict paths accurately. This analysis is a building block for more complex calculus and physical simulations.