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Chapter 5: Problem 11

Factor each polynomial. $$ x^{2}-6 x+8 $$

Short Answer

Expert verified
\((x - 4)(x - 2)\)

Step by step solution

01

- Identify coefficients

Identify the coefficients in the polynomial. Here, the polynomial is in the form of \(ax^2 + bx + c\) where \(a=1\), \(b=-6\), and \(c=8\).
02

- Find two numbers that multiply to \(c\) and add to \(b\)

Find two numbers that multiply to \(c=8\) and add to \(b=-6\). These numbers are \(-4\) and \(-2\) because \((-4)\times(-2)=8\) and \((-4)+(-2)=-6\).
03

- Write the factors

Using these numbers, write the factors of the polynomial. The polynomial \(x^2 - 6x + 8\) factors to \((x - 4)(x - 2)\).

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Key Concepts

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

Quadratic Equations
A quadratic equation is any polynomial equation of the form \(ax^2+bx+c=0\). The highest exponent in this equation is 2, making it a 'quadratic'. Quadratic equations often appear in algebra, and they can be used to describe various real-world problems. The standard form includes three terms:
  • The 'quadratic term' (\(ax^2\))
  • The 'linear term' (\(bx\))
  • The 'constant term' (\(c\))
Each coefficient plays an important role in the shape and position of the parabolic graph of the quadratic equation.
Factoring Techniques
Factoring is the process of breaking down a polynomial into simpler components called factors that, when multiplied together, give you the original polynomial. For quadratic equations, this usually involves finding two binomials. Here's a simple approach to factor a basic quadratic equation:
  • Identify the coefficients (\(a\), \(b\), and \(c\)).
  • Look for two numbers that multiply to \(c\) and add to \(b\).
  • Transform the quadratic expression into the product of two binomials.
In our given example, \(x^2 - 6x + 8\), we looked for two numbers that multiply to 8 and add to -6. The successful candidates were -4 and -2, leading us to the factored form, \((x - 4)(x - 2)\).
Polynomial Coefficients
We often describe the parts of a polynomial equation as coefficients. In the equation \(x^2 - 6x + 8\), coefficients determine the behavior and solutions of the equation. Coefficients are:
  • 'a' for the quadratic term (\(a\) in \(ax^2\))
  • 'b' for the linear term (\(b\) in \(bx\))
  • 'c' for the constant term (constant or standalone number)
Identifying these coefficients is the first step in our factoring technique. Here \(a=1\), \(b=-6\), and \(c=8\). Recognizing these values helps us apply our factoring strategy correctly.
Roots of Equations
The roots of a quadratic equation are the solutions to the equation \(ax^2 + bx + c = 0\). These are the values of \(x\) that make the equation true. The factored form of the polynomial, such as \((x - 4)(x - 2)=0\), can be easily solved to find the roots:
  • Set each factor to zero: \(x-4=0\) and \(x-2=0\);
  • Solve for \(x\): \(x=4\) and \(x=2\);
Thus, the roots of the equation \(x^2 - 6x + 8 = 0\) are \(x=4\) and \(x=2\). These roots are where the parabola representing the equation intersects the x-axis.

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Most popular questions from this chapter

Factor each polynomial and explain how you decided which method to use. a) \(x^{2}+10 x+25\) b) \(x^{2}-10 x+25\) c) \(x^{2}+26 x+25\) d) \(x^{2}-25\) e) \(x^{2}+25\)

Factor each polynomial using the trial-and-error method. $$ a^{2}+7 a+10 $$

Factor each polynomial by grouping. $$ x^{2} a^{2}+a^{2}+x^{2}+1 $$

Solve each equation for \(y .\) Assume a and b are positive numbers. $$y^{2}-b^{2}=0$$

Factor each polynomial by grouping. $$ a x+3 x+4 a+12 $$
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