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Chapter 0: Problem 29

Completely factor the polynomial. $$ 2 x^{3}-3 x^{2}+4 x-6 $$

Short Answer

Expert verified
The completely factored form of the polynomial \(2x^{3}-3x^{2}+4x-6\) is \((x - 1)(2x + 3)(x - 2)\)

Step by step solution

01

Identify Polynomial and its Degree

The given polynomial is \(2x^{3}-3x^{2}+4x-6\) and its degree is 3.
02

Find Potential Rational Roots

The potential rational roots of a polynomial can be found using the rational root theorem. According to this theorem, if a polynomial has a rational root \(p / q\) then p is a factor of the constant term and q is a factor of the leading coefficient i.e; p is a factor of -6 (卤1, 卤2, 卤3, 卤6) and q is a factor of 2 (卤1, 卤2). Therefore, the potential rational roots of the given polynomial are \(卤1, 卤2, 卤3, 卤6, 卤0.5, 卤1.5, 卤3\).
03

Substitute Potential Roots

Substitute the potential roots into the polynomial until the output equals zero. Taking \(x = 1\) we find that the polynomial equals zero. So, \(x = 1\) is a root of the polynomial.
04

Perform Polynomial Division

Divide the polynomial \(2x^{3}-3x^{2}+4x-6\) by \(x-1\) using polynomial long division. The result is \(2x^2 - x - 6\).
05

Factor the Quadratic Result

The quadratic \(2x^{2} - x - 6\) can be factored into \((2x + 3) (x - 2)\). Hence, the complete factor of the given polynomial is \((x - 1)(2x + 3)(x - 2)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Root Theorem
The Rational Root Theorem is a useful tool when we need to factor polynomials. It gives us a list of possible rational roots or zeroes for a given polynomial. These roots can be numbers that, if plugged into the polynomial, would make it equal to zero.
To apply this theorem, follow these steps:
  • Identify the constant term of the polynomial, which is the last number (here, it is -6).
  • Identify the leading coefficient, which is the coefficient of the highest degree term (here, it is 2).
  • Factors of the constant term (-6) include 卤1, 卤2, 卤3, and 卤6.
  • Factors of the leading coefficient (2) include 卤1 and 卤2.
The rational roots will be combinations of these factors, where potential roots are of the form \( \frac{p}{q} \), where \( p \) divides the constant term and \( q \) divides the leading coefficient.
This results in potential roots of 卤1, 卤2, 卤3, 卤6, 卤0.5, 卤1.5, and 卤3.
Polynomial Division
Once we have a root from the Rational Root Theorem (in this case, \( x=1 \)), we use polynomial division to simplify the polynomial further. Polynomial division is similar to long division used with numbers but is applied to polynomials.
Here's how it works:
  • Set the root (here \( x-1 \)) as the divisor.
  • Divide the original polynomial \( 2x^3 - 3x^2 + 4x - 6 \) by \( x-1 \).
  • Perform the division step-by-step, starting with the highest degree term.
  • The result will be a simplified polynomial, after which the remainder should be zero if the division was exact.
This process allows us to break down a complex polynomial into simpler expressions.
Quadratic Factoring
After polynomial division, what remains might be a quadratic expression. In this exercise, after dividing \( 2x^3 - 3x^2 + 4x - 6 \) by \( x-1 \), we get \( 2x^2 - x - 6 \).
Factoring a quadratic polynomial involves finding two binomials that multiply to give the original quadratic.
Here's the approach:
  • Look for two numbers that multiply to the product of the quadratic's leading coefficient (2) and the constant term (-6), which is -12.
  • These numbers also need to add up to the linear coefficient (-1).
  • In this case, -3 and 4 work because \(-3 \times 4 = -12\) and \(-3 + 4 = 1\).
  • Use these numbers to split the middle term, and then factor by grouping.
  • The final factors of \( 2x^2 - x - 6 \) are \((2x + 3)\) and \((x - 2)\).
By combining polynomial division and quadratic factoring, we completely factor the initial polynomial into \((x - 1)(2x + 3)(x - 2)\). This full factorization breaks the polynomial down into its simplest parts.

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Most popular questions from this chapter

find the domain of the given expression. $$ \sqrt{x^{2}+3} $$

Use the Rational Zero Theorem as an aid in finding all real zeros of the polynomial. $$ 2 x^{3}-x^{2}-13 x-6 $$

simplify the expression. $$ \frac{7 x^{2}}{x^{-3}} $$

find the domain of the given expression. $$ \sqrt{4 x^{2}+1} $$

evaluate the expression for the given value of x. $$ x-4 x^{-2} \quad x=3 $$
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