- Perform synthetic division on the coefficients \([1, 6, -7, -60]\) using \(-5\) as the root to find the quotient coefficients \(a=1, b=1, c=-12, d=0\).
- Factor the resulting quotient \(x^2 + x - 12\) into \((x - 3)(x + 4)\).
- Express the original polynomial as \(p(x) = (x+5)(x-3)(x+4)\).
- Final answers: \(a=1, b=1, c=-12, d=0\) and \(p(x) = (x+5)(x - \boxed{3})(x + \boxed{4})\).
Explanation
- Analyzing the Polynomial and Synthetic Division Process
We are given the polynomial \(p(x) = x^3 + 6x^2 - 7x - 60\) and tasked with testing whether \(-5\) is a root using synthetic division. Synthetic division is an efficient method to divide a polynomial by a linear factor of the form \((x - r)\). In this case, our divisor is \((x - (-5)) = (x + 5)\). The coefficients of our polynomial \(p(x)\) are \(1, 6, -7,\) and \(-60\).
- Performing Synthetic Division
We perform the synthetic division with the root \(-5\):
- Write the divisor \(-5\) and the coefficients \(1, 6, -7, -60\) in a row.
- Bring down the first coefficient, \(1\). This value is \(a = 1\).
- Multiply this value by the divisor: \(1 \times (-5) = -5\). Add this to the second coefficient: \(6 + (-5) = 1\). This is \(b = 1\).
- Multiply this result by the divisor: \(1 \times (-5) = -5\). Add this to the third coefficient: \(-7 + (-5) = -12\). This is \(c = -12\).
- Finally, multiply this result by the divisor: \((-12) \times (-5) = 60\). Add this to the last coefficient: \(-60 + 60 = 0\). This is \(d = 0\).
Since the remainder \(d\) is \(0\), \(-5\) is indeed a root of the polynomial. The quotient from this division is \(1x^2 + 1x - 12\).
- Factoring the Quotient and Finalizing the Polynomial Form
Now that we have the quotient \(x^2 + x - 12\), we factor it to find the remaining roots. We look for two numbers that multiply to \(-12\) and add to \(1\). These numbers are \(4\) and \(-3\). Thus, the quadratic factors as:
\[x^2 + x - 12 = (x + 4)(x - 3)\]
Substituting this back into our expression for \(p(x)\), we get:
\[p(x) = (x + 5)(x + 4)(x - 3)\]
Following the required format \((x+5)(x - \square)(x + \square)\), we rearrange this as:
\[p(x) = (x + 5)(x - 3)(x + 4)\]
Thus, the empty boxes are \(3\) and \(4\).
Examples
Synthetic division is widely used in computer science for evaluating polynomials efficiently at specific values. For example, in computer graphics, when calculating curves or surfaces defined by cubic splines, algorithms must frequently divide polynomials to transform or simplify coordinate equations. Mastering this technique allows for faster computation of complex geometric shapes, leading to smoother animations and more efficient rendering processes in video game engines.