- Perform synthetic division on the polynomial \(x^3 - 5x^2 - 7x - 42\) using the root \(-4\).
- Identify the coefficients of the resulting quotient as \(1, -9, 29\).
- Express the quotient in standard form: \(x^2 - 9x + 29\).
- Conclude that the second term of the quotient is \(\boxed{-9x}\).
Explanation
- Analyzing the Problem and Setting Up Synthetic Division
We are dividing the polynomial \(P(x) = x^3 - 5x^2 - 7x - 42\) by the linear factor \(D(x) = x + 4\). To find the quotient, we use the method of synthetic division. Synthetic division is a shortcut for polynomial long division when the divisor is in the form \(x - k\). Here, since we are dividing by \(x + 4\), we have \(k = -4\). The coefficients of our dividend \(P(x)\) are \(1, -5, -7,\) and \(-42\).
- Performing Synthetic Division
To perform the synthetic division, we follow these steps:
- Write down the coefficients of the polynomial: \(1, -5, -7, -42\).
- Use the root \(-4\) as the multiplier.
- Bring down the first coefficient, \(1\).
- Multiply this value by the root: \(1 \times (-4) = -4\). Add this to the second coefficient: \(-5 + (-4) = -9\).
- Multiply this result by the root: \(-9 \times (-4) = 36\). Add this to the third coefficient: \(-7 + 36 = 29\).
- Multiply this result by the root: \(29 \times (-4) = -116\). Add this to the constant term: \(-42 + (-116) = -158\).
The resulting values from the synthetic division are the coefficients of the quotient and the remainder. The quotient coefficients are \(1, -9, 29\), and the remainder is \(-158\).
- Determining the Second Term of the Quotient
From the synthetic division coefficients \([1, -9, 29]\), we can write the resulting quotient polynomial as:
\[Q(x) = 1x^2 - 9x + 29\]
The quotient is a quadratic polynomial in standard form \(ax^2 + bx + c\). In this form, the first term is \(x^2\), the second term is \(-9x\), and the third term is \(29\). The second term of the quotient is therefore \(-9x\). The final value for the second term is \(-9x\).
Examples
In computer programming, synthetic division is often used to efficiently evaluate polynomials or to perform root finding algorithms. By dividing a polynomial, one can reduce its degree, making it simpler to find remaining roots. This is fundamental in tasks like signal processing, where polynomials describe the frequency response of filters, and in graphics, where they define curves and surfaces. Understanding how to divide polynomials helps in optimizing these calculations, ensuring software runs faster and more accurately.