8.1: Diagrams of Products
- Explain why the diagram shows that \(6(3+4) = 6 \boldcdot 3 + 6 \boldcdot 4\).
- Draw a diagram to show that \(5(x+2) = 5x + 10\).
8.2: Drawing Diagrams to Represent More Products
Applying the distributive property to multiply out the factors of, or expand, \(4(x+2)\) gives us \(4x + 8\), so we know the two expressions are equivalent. We can use a rectangle with side lengths \((x+2)\) and 4 to illustrate the multiplication.
- Draw a diagram to show that \(n(2n+5)\) and \(2n^2 + 5n\) are equivalent expressions.
- For each expression, use the distributive property to write an equivalent expression. If you get stuck, consider drawing a diagram.
a. \(6\left(\frac13 n + 2\right)\)
b. \(p(4p + 9)\)
c. \(5r\left(r + \frac35\right)\)
d. \((0.5w + 7)w\)
8.3: Using Diagrams to Find Equivalent Quadratic Expressions
- Here is a diagram of a rectangle with side lengths \(x+1\) and \(x+3\). Use this diagram to show that \((x+1)(x+3)\) and \(x^2 + 4x+3\) are equivalent expressions.
- Draw diagrams to help you write an equivalent expression for each of the following:
- Write an equivalent expression for each expression without drawing a diagram:
- \((x +2)(x + 6)\)
- \((x +5)(2x + 10)\)
- Is it possible to arrange an \(x\) by \(x\) square, five \(x\) by 1 rectangles and six 1 by 1 squares into a single large rectangle? Explain or show your reasoning.
- What does this tell you about an equivalent expression for \(x^2 + 5x + 6\)?
- Is there a different non-zero number of 1 by 1 squares that we could have used instead that would allow us to arrange the combined figures into a single large rectangle?
A quadratic function can often be defined by many different but equivalent expressions. For example, we saw earlier that the predicted revenue, in thousands of dollars, from selling a downloadable movie at \(x\) dollars can be expressed with \(x(18-x)\), which can also be written as \(18x - x^2\). The former is a product of \(x\) and \(18-x\), and the latter is a difference of \(18x\) and \(x^2\), but both expressions represent the same function.
Sometimes a quadratic expression is a product of two factors that are each a linear expression, for example \((x+2)(x+3)\). We can write an equivalent expression by thinking about each factor, the \((x+2)\) and \((x+3)\), as the side lengths of a rectangle, and each side length decomposed into a variable expression and a number.
Multiplying \((x+2)\) and \((x+3)\) gives the area of the rectangle. Adding the areas of the four sub-rectangles also gives the area of the rectangle. This means that \((x+2)(x+3)\) is equivalent to \(x^2 + 2x + 3x + 6\), or to \(x^2 + 5x + 6\).
Notice that the diagram illustrates the distributive property being applied. Each term of one factor (say, the \(x\) and the 2 in \(x+2\)) is multiplied by every term in the other factor (the \(x\) and the 3 in \(x+3\)).
In general, when a quadratic expression is written in the form of \((x+p)(x+q)\), we can apply the distributive property to rewrite it as \(x^2 + px + qx + pq\) or \(x^2 + (p+q)x + pq\).