common factor

Examples of common factor in the following topics:

• Introduction to Factoring Polynomials

  • Factoring by grouping divides the terms in a polynomial into groups, which can be factored using the greatest common factor.
  • Factor out the greatest common factor, $4x(x+5) + 3y(x+5)$.
  • Factor out the binomial $(x+5)(4x+3y)$.
  • One way to factor polynomials is factoring by grouping.
  • Both groups share the same factor $(x+5)$, so the polynomial is factored as:

• Finding Factors of Polynomials

  • We now divide each term with this common factor to fill in the blanks.
  • The common factor is $3$.
  • Whenever trying to factor a complicated expression, always begin by looking for common factors that can be pulled out.
  • The factor must be common to all the terms.
  • This follows the rule: always begin by pulling out common factors before trying anything else.

• Fractions

  • Find a common denominator, and change each fraction to an equivalent fraction using that common denominator.
  • If any numerator and denominator shares a common factor, the fractions can be reduced to lowest terms before or after multiplying.
  • For example, the resulting fraction from above can be reduced to $\frac{1}{2}$ because the numerator and denominator share a factor of  6.
  • Alternatively, the fractions in the initial equation could have been reduced, as shown below, because 2 and 4 share a common factor of 2 and 3 and 3 share a common factor of 3:
  • A common situation where multiplying fractions comes in handy is during cooking.

• Rational Algebraic Expressions

  • But how do you find the least common denominator?
  • We start, as usual, by factoring.
  • When we add or subtract rational expressions, we will not simply be considering the prime factors of integers when looking for the least common denominator.
  • Rather, we will be looking for monomial and binomial factors that are common to both rational expressions.
  • Finding the prime factors of the denominators of two fractions enables us to find a common denominator.

• Simplifying, Multiplying, and Dividing

  • Performing these operations on rational expressions often involves factoring polynomial expressions out of the numerator and denominator.
  • Rational expressions can be simplified by factoring the numerator and denominator where possible, and canceling terms.
  • This can be simplified by canceling out one factor of $x$ in the numerator and denominator, which gives the expression $3x^2$.
  • which, after canceling the common factor of $(x+2)$ from both the numerator and denominator, gives the simplified expression

• Solving Equations with Rational Expressions; Problems Involving Proportions

  • When given the rational equation: $\displaystyle\frac a b=\frac c d$ This can be solved by either finding a common denominator, or by setting it up like: $ad=cb$ and then solving it algebraically.
  • This in turn suggests a strategy: find a common denominator, and then set the numerators equal.
  • by factoring the denominators,we find that we must multiply the left side of the equation by $\displaystyle \frac {x(x-2)}{x(x-2)}$ and the right side of the equation by $\displaystyle \frac {x+6}{x+6}$ , giving
  • and then factor.
  • A common mistake in this kind of problem is to divide both sides by $x$; this loses one of the two solutions.

• Completing the Square

  • Along with factoring and using the quadratic formula, completing the square is a common method for solving quadratic equations.  
  • It is often implemented when factoring is not an option, such as when the quadratic is a not already a perfect square.

• Factors

  • This is a complete list of the factors of 24.
  • Therefore, 2 and 3 are prime factors of 6.
  • However, 6 is not a prime factor.
  • To factor larger numbers, it can be helpful to draw a factor tree.
  • This factor tree shows the factorization of 864.

• Introduction to Ellipses

  • Recall that a circle is defined as the set of all points equidistant from a common center.
  • To do this, we introduce a scaling factor into one or both of the x-y coordinates.
  • Let's start by dividing all x coordinates by a factor $a$, and therefore scaling the x values.
  • Similarly, we can scale all the y-values by a factor $b$ (we also assume $b > 1$).
  • Connect the equation for an ellipse to the equation for a circle with stretching factors

• Summing the First n Terms in a Geometric Sequence

  • By utilizing the common ratio and the first term of a geometric sequence, we can sum its terms.
  • The following are several geometric series with different common ratios.
  • The behavior of the terms depends on the common ratio $r$:
  • where $a$ is the first term of the series, and $r$ is the common ratio.
  • Also, note that $r = 3$, because each term is multiplied by a factor of 3 to find the subsequent term.