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Magazine Chapter 4: Problem 40
Use synthetic division to divide the first polymomial by the second. $$x^{3}-2 x+1 \quad x+4$$
Short Answer
Expert verified
The quotient is \(x^2 - 4x + 14\) with remainder \(-55\).
Step by step solution
01
Identify the Divisor and the Dividend
The first polynomial \(x^3 - 2x + 1\) is the dividend, and the polynomial \(x + 4\) is the divisor. In synthetic division, we use the root of the divisor (\(x + 4 = 0\) gives \(x = -4\)) as the value for our synthetic division.
02
Set Up Synthetic Division
Write down the coefficients of the dividend \(x^3 - 2x + 1\), which are \(1, 0, -2, 1\). The coefficient for \(x^2\) is zero because it is missing from the polynomial. Now, set these coefficients up in a row with \(-4\) to the left.
03
Start the Synthetic Division
Bring down the first coefficient (\(1\)) to the bottom row. Multiply this coefficient by \(-4\) (which is the root of the divisor) and write the result under the second coefficient \(0\).
04
Continue the Division Process
Add the numbers in the second column (\(0 + (-4) = -4\)). Write \(-4\) in the third row. Multiply \(-4\) by \(-4\), which results in \(16\). Write \(16\) under the third column \(-2\).
05
Complete the Division
Add the numbers in the third column: \(-2 + 16 = 14\). Write \(14\) in the third row. Multiply \(14\) by \(-4\), which gives \(-56\). Write \(-56\) under the last column \(1\).
06
Finalize the Remainder
Add the numbers in the last column (\(1 + (-56) = -55\)). Write \(-55\) in the remainder position. The bottom row now shows the coefficients of the quotient and the remainder.
07
Interpret the Result
The result of the division is \(x^2 - 4x + 14\) with a remainder of \(-55\). Therefore, \(x^3 - 2x + 1\) divided by \(x + 4\) equals \(x^2 - 4x + 14 - \frac{55}{x+4}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Polynomials
Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. Essentially, these are combinations of different powers of the same variable, like in the polynomial \(x^3 - 2x + 1\). Here, the powers of \(x\) are 3, 1, and 0 with coefficients 1, -2, and 1, respectively. The term "polynomial" comes from "poly," meaning many, and "nomial," referring to terms, so a polynomial literally has "many terms."
- Standard form: A polynomial is usually written in descending order of powers, starting from the highest power of the variable down to the lowest.
- Degree of a polynomial: The degree is the highest power of the variable, which in our example is 3.
- Constant term: A polynomial may include a term with zero power, which is just a constant.
Exploring the Roots of Polynomials
Roots of polynomials are the values of the variable that make the polynomial equal to zero. If a polynomial is 'solved,' finding these roots means identifying where the expression equals zero when plotted on a graph. Consider a simple polynomial like \(x + 4\): setting this equal to zero gives a root of \(-4\). This is a crucial step in synthetic division, as we use this root in the process.
- Real roots: Values that are actual numbers on the real number line.
- Complex roots: These include imaginary numbers and are not situated on the real number line.
- Multiplicity of roots: A single root might be repeated more than once, affecting polynomial behavior.
The Remainder Theorem in Synthetic Division
The Remainder Theorem provides a quick way to find remainders when dividing a polynomial by a linear divisor. It states that for a polynomial \(f(x)\) and a linear divisor of \(x - c\), the remainder of the division is \(f(c)\). In synthetic division, particularly, this is evident at the last step where the remainder is visible. For \(x^3 - 2x + 1\) divided by \(x + 4\), substituting the root \(-4\) into the polynomial gives the remainder directly as \(-55\).
- Quick checks: Evaluate the function for the divisor's root to confirm the remainder.
- Polynomial evaluation: The theorem allows the remainder to quickly infer polynomial values at specific points.
- Simplifying calculations: Particularly helpful for lower-degree divisors or when polynomial evaluations are cumbersome.
