Problem 16
Question
Find the quotient and remainder using long division. $$\frac{x^{3}-x^{2}-2 x+6}{x-2}$$
Step-by-Step Solution
Verified Answer
Quotient: \(x^2 + x\), Remainder: 10.
1Step 1: Setup the Long Division
To start, set up the long division problem by writing the dividend \(x^3 - x^2 - 2x + 6\) below the division bar and the divisor \(x - 2\) outside the division bar.
2Step 2: Divide the First Terms
Divide the first term of the dividend \(x^3\) by the first term of the divisor \(x\): \(\frac{x^3}{x} = x^2\). Write \(x^2\) above the division bar.
3Step 3: Multiply and Subtract
Multiply \(x^2\) by the entire divisor \(x - 2\) to get \(x^3 - 2x^2\). Subtract this from the original dividend, which gives \((x^3 - x^2 - 2x + 6) - (x^3 - 2x^2) = x^2 - 2x + 6\).
4Step 4: Bring Down and Divide Again
Bring down the next term in the dividend, which is \(-2x\), to have \(x^2 - 2x\). Now divide \(x^2\) by \(x\) to get \(x\), and write \(x\) above the division bar next to \(x^2\).
5Step 5: Multiply and Subtract Again
Multiply \(x\) by the divisor \(x - 2\), resulting in \(x^2 - 2x\), and subtract it from \(x^2 - 2x\), yielding \(0 + 6 = 6\).
6Step 6: Final Division and Subtraction
Bring down the final term \(+6\). Since the remainder \(6\) is less than the degree of \(x\), we have our remainder. The division yields the quotient \(x^2 + x\) and remainder \(10\), as we got \(x\) term's multiplication result \((x - 2)\) that wasn't zero.
Key Concepts
Remainder TheoremQuotient of PolynomialsDegree of Polynomial
Remainder Theorem
The Remainder Theorem is a simple yet powerful tool in polynomial algebra. According to this theorem, if you divide a polynomial \(f(x)\) by a linear divisor of the form \(x - c\), the remainder of the division is simply \(f(c)\). This means you only need to evaluate the polynomial at \(c\) to find the remainder.
For example, in our problem, we divided \(x^3 - x^2 - 2x + 6\) by \(x - 2\). According to the theorem, if you plug \(x = 2\) into the polynomial, the result will be the remainder. Calculating, \(2^3 - 2^2 - 2(2) + 6 = 8 - 4 - 4 + 6 = 6\). Therefore, the remainder of our division is \(6\). This confirms what was found through long division.
Understanding the Remainder Theorem not only helps in verifying division results quickly but also provides insight into the behavior of polynomials.
For example, in our problem, we divided \(x^3 - x^2 - 2x + 6\) by \(x - 2\). According to the theorem, if you plug \(x = 2\) into the polynomial, the result will be the remainder. Calculating, \(2^3 - 2^2 - 2(2) + 6 = 8 - 4 - 4 + 6 = 6\). Therefore, the remainder of our division is \(6\). This confirms what was found through long division.
Understanding the Remainder Theorem not only helps in verifying division results quickly but also provides insight into the behavior of polynomials.
Quotient of Polynomials
The quotient of polynomials refers to the result you get when you divide one polynomial by another. Just like regular numbers, when you divide polynomials, you may not always get another polynomial without a remainder.
In the provided exercise, dividing \(x^3 - x^2 - 2x + 6\) by \(x - 2\) gave us the quotient \(x^2 + x\). This is what you would multiply back by the divisor to get as close as possible to the original dividend.
Understanding quotients is essential because it helps in simplifying expressions and solving polynomial equations.
Here's a summary of steps involved:
In the provided exercise, dividing \(x^3 - x^2 - 2x + 6\) by \(x - 2\) gave us the quotient \(x^2 + x\). This is what you would multiply back by the divisor to get as close as possible to the original dividend.
Understanding quotients is essential because it helps in simplifying expressions and solving polynomial equations.
Here's a summary of steps involved:
- Identify the highest degree term of the dividend and the divisor.
- Divide the leading term of the dividend by the leading term of the divisor.
- Subtract the resulting expression from the original polynomial.
- Repeat the process with the new polynomial formed by subtraction.
Degree of Polynomial
The degree of a polynomial is a very crucial concept to grasp. It refers to the highest power of the variable in the polynomial expression. This concept helps in understanding the behavior and the graph of the polynomial.
For instance, the polynomial \(x^3 - x^2 - 2x + 6\) has a degree of three because the highest power of \(x\) is three. When dividing this polynomial by \(x - 2\), which has a degree of one, the degree of the resulting quotient (\(x^2 + x\)) is two. This decrease in degree occurs because we are effectively reducing polynomial terms through division.
Degrees are also significant for factors. The remainder left after a division will have a degree that is less than the divisor. In this case, since our divisor is \(x - 2\), the remainder is a constant, \(6\) in our problem, which has a degree of zero.
Degrees help not only in performing polynomial divisions but also in estimating roots and sketching graphs.
For instance, the polynomial \(x^3 - x^2 - 2x + 6\) has a degree of three because the highest power of \(x\) is three. When dividing this polynomial by \(x - 2\), which has a degree of one, the degree of the resulting quotient (\(x^2 + x\)) is two. This decrease in degree occurs because we are effectively reducing polynomial terms through division.
Degrees are also significant for factors. The remainder left after a division will have a degree that is less than the divisor. In this case, since our divisor is \(x - 2\), the remainder is a constant, \(6\) in our problem, which has a degree of zero.
Degrees help not only in performing polynomial divisions but also in estimating roots and sketching graphs.
Other exercises in this chapter
Problem 16
Evaluate the expression and write the result in the form \(a+b i\) $$(2+5 i)+(4-6 i)$$
View solution Problem 16
Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=x^{3}-7 x^{2}+14 x-8$$
View solution Problem 16
A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its \(x\) - and \(y\) -intercept(s). (c) Sketch its
View solution Problem 16
Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. (GRAPH CANT COPY) $$P(x)=(x-1)(x+1)
View solution