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Chapter 3: Polynomials

    If you don’t want to memorize the Pascal’s Triangle, there is another way to expand the polynomials. For example: if you want to expand (a + b)³, the you put the first variable a to a³; then you move the exponent of the first expanded term to the coefficient of the second term and decrease the exponent of the first variable by one and increase the second by one. Now you get a³ + 3a²b. Then you multiply the coefficient of the second term and the exponent of the first variable and divide the result by the placing number of the expanded term. Now you get 3 x 2 / 2 = 3(the term is in the second place). The result you get is the coefficient of the third expand term. As usual, you decrease the exponent of the first variable by one and increase the second by one. You can use this way to find all the expanded terms. Just note that the number of terms of an expanded polynomial equals the exponent of the non-expanded polynomial + 1.

    When you need to un-expand or factor the expanded the polynomials, you can use the Pascal’s Triangle too. But there is also another way if you don’t want to memorize the triangle. This way involves finding the factors of the polynomials. The factors of the polynomials are the numbers that can be substituted into the variables of the polynomials and evaluate the equation to zero. When you factor a polynomial, you take out each coefficient of each term in the polynomial. Then you line them up. After that, you need to find call the positive and negative factors of the first and last coefficient. When you are done, you need to list them out on the scratch paper. Then you start try each number into the row of the coefficients. You take the first coefficient down and multiply with the number and then add to the second coefficient. You then do the same thing with the following coefficients. If the result you get after adding the last coefficient is zero, then the number is a factor of the polynomial. If not, you need to try additional numbers. After all the coefficient but the first one has turned into zeros, you have found all the factors of the polynomial. Then you just put each factor in the form of variable – the factor with bracket around them, and multiply them together. Here are some additional ways to factor the polynomials:

  a² + b² = (a + b)(a – b)

  a³ + b³ = (a + b)(a² – ab + b²)

  a³ - b³ = (a - b)(a² + ab + b²)

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