March 30, 2010 Using Factorization in Geometry
Chapter 8 Reteach #2 ‐ Answers 1. The rectangle shown has an area of (x2 – 2x – 24) m2. What expression would represent the length of the missing side? What would the rectangle’s perimeter be if x were to equal 9? (x + 4) m ? Since we know that one of the sides is (x + 4) m, we know that if you factor x2 – 2x + 24 you’ll have to get (x + 4) and the missing side. So let’s factor x2 – 2x + 24: What two numbers multiply to give –24 x2 – 2x – 24 (x + 4)(x – 6) and add to give –2? x2 – 6x + 4x – 24 Check (FOIL) D x2 – 2x – 24 So the expression that represents the missing side is (x – 6) m. If x were to equal 9, the rectangle would look like this: (9 + 4) = 13 m (9 – 6) = 3 m So the perimeter of the rectangle would be 13 + 13 + 3 + 3 = 32 m. 2. The area of yet another rectangle is (x2 + 10x + 16) in2. The length and the width of the rectangle can be written in the form x + b, where b is a whole number. Is it possible that the rectangle in this problem could be a square? How do you know? If the answer is no, rewrite the trinomial so that the rectangle would be a square. If the rectangle is a square, the two sides will have to be the same. (In other words, if one side were (x + 4) inches, the other side would also have to be (x + 4) inches. The question, then is really asking whether x2 + 10x + 16 is a perfect‐square trinomial. Factoring x2 + 10x + 16 gives us (x + 8)(x + 2), so the answer to the question is no – the rectangle is not a square. Rewriting the trinomial so that it would be a square just means coming up with a perfect‐square trinomial. The most obvious is: x2 + 8x + 16 What two numbers multiply to give 16 (x + 4)(x + 4) and add to give 8? 2
Perfect‐square trinomial (x + 4) x2 + 4x + 4x + 16 Check (FOIL) D x2 + 8x + 16 So the trinomial x2 + 8x + 16 would represent a rectangle that is square. Any perfect‐square trinomial where what’s being squared matches the form (x + b) would be a correct answer, not just (x + 4)2, though. So all of the following would have been correct answers: … which is (x + 1)2 x2 + 2x + 1 … which is (x + 2)2 x2 + 4x + 4 … which is (x + 3)2 x2 + 6x + 9 … and so on. 3. Tim constructs yet another stupid rectangular fence just to mess with your minds. The enclosed space has an area of (3x2 + 7x + 2) square feet. The length and the width of the enclosed garden space can be written in the form ax + b, where a and b are both whole numbers. What expressions would represent the length and the width of the enclosed garden space? What would be the area and perimeter of the space when x = 3? The problem asks us again for the sides of a rectangle, and we know the sides will have the form ax + b. So we’ll need to factor 3x2 + 7x + 2. Let’s do that: 3x2 + 7x + 2 Factors of 3 are 3 and 1, Factors of 2 are 2 and 1 (3x + 1)(x + 2) Factored 3x2 + 6x + x + 3 3x2 + 7x + 2 Check (FOIL) D So the expressions that represent the length and the width of the garden space would be (3x + 1) feet and (x + 2) feet. 3x + 1 x + 2 If that’s true, if we say that x = 3, that gives us a length and width of 3 ∙ 3 + 1 = 10 feet and 3 + 2 = 5 feet. (3 ∙ 3 + 1) = 10 feet (3 + 2) = 5 feet So our area would be 10 ∙ 5 = 50 square feet. And our perimeter would be 10 + 10 + 5 + 5 = 30 feet.