- Divide the polynomial \(x^3 - 7x^2 - 16x + 112\) by \((x - 4)\) to get the quadratic factor \(x^2 - 3x - 28\).
- Factor the quadratic expression into \((x - 7)(x + 4)\).
- Assemble the factors: \((x - 4)(x - 7)(x + 4)\).
- Order the factors from smallest constant to largest: \((x - 7)(x - 4)(x + 4)\) - wait, checking order: constants are -7, -4, 4. So \((x-7)(x-4)(x+4)\).
- The final result is \(\boxed{(x-7)(x-4)(x+4)}\).
Explanation
- Analysis of the Polynomial Function
We are given the polynomial function \(f(x) = x^3 - 7x^2 - 16x + 112\) and we are told that \(4\) is one of its zeros. This implies that \((x - 4)\) is a factor of the polynomial. Our objective is to find the fully factored form of \(f(x)\). A third-degree polynomial has three linear factors. Since we already know one factor, \((x - 4)\), we can use division to find the remaining quadratic factor.
- Dividing the Polynomial
We perform polynomial division by dividing \(f(x) = x^3 - 7x^2 - 16x + 112\) by \((x - 4)\).
- Dividing \(x^3\) by \(x\) gives \(x^2\). Multiplying \(x^2\) by \((x - 4)\) gives \(x^3 - 4x^2\). Subtracting this from \(f(x)\) leaves \(-3x^2 - 16x + 112\).
- Dividing \(-3x^2\) by \(x\) gives \(-3x\). Multiplying \(-3x\) by \((x - 4)\) gives \(-3x^2 + 12x\). Subtracting this from \(-3x^2 - 16x + 112\) leaves \(-28x + 112\).
- Dividing \(-28x\) by \(x\) gives \(-28\). Multiplying \(-28\) by \((x - 4)\) gives \(-28x + 112\). The remainder is \(0\).
Thus, the quotient is \(x^2 - 3x - 28\). The polynomial can be written as \(f(x) = (x - 4)(x^2 - 3x - 28)\).
- Factoring the Quadratic and Ordering Factors
Now we factor the quadratic expression \(x^2 - 3x - 28\). We look for two numbers that multiply to \(-28\) and add to \(-3\). These numbers are \(-7\) and \(4\), since \((-7) \times 4 = -28\) and \((-7) + 4 = -3\). Thus, the quadratic factors as \((x - 7)(x + 4)\). The full factorization is \(f(x) = (x - 4)(x - 7)(x + 4)\). To order the factors from smallest to largest based on the constant terms (or roots), we examine them:
- Roots are \(4, 7, -4\).
- Ordered from smallest to largest, the roots are \(-4, 4, 7\).
- The corresponding factors are \((x + 4), (x - 4), (x - 7)\).
Examples
Factoring polynomials is a fundamental skill in algebra with many real-world applications. For example, engineers use it to solve differential equations that describe physical systems like bridge vibrations or electrical circuit behavior. By breaking a complex polynomial down into its simple linear factors, they can identify the specific frequencies or values at which a system reaches equilibrium or instability. This makes it possible to design safer infrastructure and more efficient electronic components.