Lesson 7

Rewriting Quadratic Expressions in Factored Form (Part 2)

  • Let’s write some more expressions in factored form.

7.1: Sums and Products

  1. The product of the integers 2 and -6 is -12. List all the other pairs of integers whose product is -12.
  2. Of the pairs of factors you found, list all pairs that have a positive sum. Explain why they all have a positive sum.
  3. Of the pairs of factors you found, list all pairs that have a negative sum. Explain why they all have a negative sum.

7.2: Negative Constant Terms

  1. These expressions are like the ones we have seen before.

    factored form standard form
    \((x+5)(x+6)\)  
      \(x^2+13x+30\)
    \((x-3)(x-6)\)  
      \(x^2-11x+18\)

    Each row has a pair of equivalent expressions.

    Complete the table. If you get stuck, consider drawing a diagram.

  2. These expressions are in some ways unlike the ones we have seen before.

    factored form standard form
    \((x+12)(x-3)\)  
      \(x^2-9x-36\)
      \(x^2-35x-36\)
      \(x^2+35x-36\)

    Each row has a pair of equivalent expressions.

    Complete the table. If you get stuck, consider drawing a diagram.

  3. Name some ways that the expressions in the second table are different from those in the first table (aside from the fact that the expressions use different numbers).

7.3: Factors of 100 and -100

  1. Consider the expression \(x^2 + bx +100\).

    Complete the first table with all pairs of factors of 100 that would give positive values of \(b\), and the second table with factors that would give negative values of \(b\).

    For each pair, state the \(b\) value they produce. (Use as many rows as needed.)

    positive value of \(b\)

    factor 1 factor 2 \(b\) (positive)
         
         
         
         
         

    negative value of \(b\)

    factor 1 factor 2 \(b\) (negative)
         
         
         
         
         
  2. Consider the expression \(x^2 + bx -100\).

    Complete the first table with all pairs of factors of -100 that would result in positive values of \(b\), the second table with factors that would result in negative values of \(b\), and the third table with factors that would result in a zero value of \(b\).

    For each pair of factors, state the \(b\) value they produce. (Use as many rows as there are pairs of factors. You may not need all the rows.)

    positive value of \(b\)

    factor 1 factor 2 \(b\) (positive)
         
         
         
         
         

    negative value of \(b\)

    factor 1 factor 2 \(b\) (negative)
         
         
         
         
         

    zero value of \(b\)

    factor 1 factor 2 \(b\) (zero)
         
         
         

     

  3. Write each expression in factored form:

    1. \(x^2 - 25x + 100\)
    2. \(x^2 + 15x - 100\)
    3. \(x^2 - 15x - 100\)
    4. \(x^2 + 99x - 100\)


How many different integers \(b\) can you find so that the expression \(x^2+10x+b\) can be written in factored form?

Summary

When we rewrite expressions in factored form, it is helpful to remember that:

  • Multiplying two positive numbers or two negative numbers results in a positive product.
  • Multiplying a positive number and a negative number results in a negative product.

This means that if we want to find two factors whose product is 10, the factors must be both positive or both negative. If we want to find two factors whose product is -10, one of the factors must be positive and the other negative.

Suppose we wanted to rewrite \(x^2 -8x + 7\) in factored form. Recall that subtracting a number can be thought of as adding the opposite of that number, so that expression can also be written as \(x^2 + \text-8x + 7\). We are looking for two numbers that:

  • Have a product of 7. The candidates are 7 and 1, and -7 and -1.
  • Have a sum of -8. Only -7 and -1 from the list of candidates meet this condition.

The factored form of \(x^2 -8x + 7\) is therefore \((x + \text-7)(x + \text-1)\) or, written another way, \((x-7)(x-1)\).

To write \(x^2 + 6x - 7\) in factored form, we would need two numbers that:

  • Multiply to make -7. The candidates are 7 and -1, and -7 and 1.
  • Add up to 6. Only 7 and -1 from the list of candidates add up to 6.

The factored form of \(x^2 + 6x - 7\) is \((x+7)(x-1)\).

Glossary Entries

  • coefficient

    In an algebraic expression, the coefficient of a variable is the constant the variable is multiplied by. If the variable appears by itself then it is regarded as being multiplied by 1 and the coefficient is 1.

    The coefficient of \(x\) in the expression \(3x + 2\) is \(3\). The coefficient of \(p\) in the expression \(5 + p\) is 1.

  • constant term

    In an expression like \(5x + 2\) the number 2 is called the constant term because it doesn't change when \(x\) changes. 

    In the expression \(5x-8\) the constant term is -8, because we think of the expression as \(5x + (\text-8)\). In the expression \(12x-4\) the constant term is -4.

  • linear term

    The linear term in a quadratic expression (In standard form) \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, is the term \(bx\). (If the expression is not in standard form, it may need to be rewritten in standard form first.)

  • zero product property

    The zero product property says that if the product of two numbers is 0, then one of the numbers must be 0.