An Intro to Factoring

\pi \cdot \pi

An Intro to Factoring

  • A factor is a number that divides another number completely, with no remainder.

What is a Factor?

  • We say 2 is a factor of 8 because \(\frac{8}{2}=4\) with a remainder of 0.
  • 2 is also a factor of 12 because \(\frac{12}{2}=6\) with no remainder.
  • However, 2 is not a factor of 7 because when we divided 7 by 2, we get a quotient of 3 and a remainder of 1.

An Intro to Factoring

  • A factor is a number that divides another number completely, with no remainder.

What is a Factor?

  • We say 2 is a factor of 8 because \(\frac{8}{2}=4\) with a remainder of 0.
  • 2 is also a factor of 12 because \(\frac{12}{2}=6\) with no remainder.
  • However, 2 is not a factor of 7 because when we divided 7 by 2, we get a quotient of 3 and a remainder of 1.

An Intro to Factoring

  • A factor is a number that divides another number completely, with no remainder.

What is a Factor?

  • We say 2 is a factor of 8 because \(\frac{8}{2}=4\) with a remainder of 0.
  • 2 is also a factor of 12 because \(\frac{12}{2}=6\) with no remainder.
  • However, 2 is not a factor of 7 because when we divided 7 by 2, we get a quotient of 3 and a remainder of 1.

An Intro to Factoring

  • A factor is a number that divides another number completely, with no remainder.

What is a Factor?

  • We say 2 is a factor of 8 because \(\frac{8}{2}=4\) with a remainder of 0.
  • 2 is also a factor of 12 because \(\frac{12}{2}=6\) with no remainder.
  • However, 2 is not a factor of 7 because when we divided 7 by 2, we get a quotient of 3 and a remainder of 1.

An Intro to Factoring

Two Sides of the Same Coin

  • Multiplication
  • We can say that 35 is a multiple of 5.
  • Also, we can say that 35 is a multiple of 7.

\((5) \cdot (7) = 35\)

  • Factorization
  • We can say that 5 is a factor of 35 because \(\frac{35}{5}=7\).
  • Also, we can say that 7 is a factor of 35 because \(\frac{35}{7}=5\).

\(35=(5) \cdot (7)\)

An Intro to Factoring

Two Sides of the Same Coin

  • Multiplication
  • We can say that 35 is a multiple of 5.
  • Also, we can say that 35 is a multiple of 7.

\((5) \cdot (7) = 35\)

  • Factorization
  • We can say that 5 is a factor of 35 because \(\frac{35}{5}=7\).
  • Also, we can say that 7 is a factor of 35 because \(\frac{35}{7}=5\).

\(35=(5) \cdot (7)\)

An Intro to Factoring

Two Sides of the Same Coin

  • Multiplication
  • We can say that 35 is a multiple of 5.
  • Also, we can say that 35 is a multiple of 7.

\((5) \cdot (7) = 35\)

  • Factorization
  • We can say that 5 is a factor of 35 because \(\frac{35}{5}=7\).
  • Also, we can say that 7 is a factor of 35 because \(\frac{35}{7}=5\).

\(35=(5) \cdot (7)\)

An Intro to Factoring

Two Sides of a Similar Coin

Multiplication

  • We say that \((x^2+3x+2)\) is a multiple of \((x+1)\) and \((x+2)\).
  • We say that \((x+1)\) and \((x+2)\) are factors of \((x^2+3x+2)\).

\((x+1) \cdot (x+2) = x^2+3x+2\)

Factorization

An Intro to Factoring

Two Sides of a Similar Coin

Multiplication

  • We say that \((x^2+3x+2)\) is a multiple of \((x+1)\) and \((x+2)\).
  • We say that \((x+1)\) and \((x+2)\) are factors of \((x^2+3x+2)\).

\((x+1) \cdot (x+2) = x^2+3x+2\)

Factorization

An Intro to Factoring

Two Sides of a Similar Coin

Multiplication

  • We say that \((x^2+3x+2)\) is a multiple of \((x+1)\) and \((x+2)\).
  • We say that \((x+1)\) and \((x+2)\) are factors of
    \((x^2+3x+2)\).

\((x+1) \cdot (x+2) = x^2+3x+2\)

Factorization

An Intro to Factoring

Looking ahead...

For the majority of the time we have left this semester, we will solve problems that will, at one time or another, require us to factor. In this chapter, we will learn how to factor various problems.

 

For instance, when we need to factor \(25x^2+20xy+4y^2\), we can ask, "what would we multiply to get \(25x^2+20xy+4y^2\)?"

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