Polynomial Factoring B43C83

1. **Problem:** A square garden has an area of $$x^2 + 10x + 25$$ square meters. Find the side length by expressing the area as the product of two equal factors. 2. **Formula:** The area of a square is $$side \times side = side^2$$. 3. **Step 1:** Recognize the polynomial as a perfect square trinomial. $$x^2 + 10x + 25 = (x + 5)^2$$ 4. **Step 2:** Therefore, the side length is $$x + 5$$ meters. --- 1. **Problem:** A square floor has an area of $$4m^2 - 20m + 25$$ square meters. Find the side length. 2. **Step 1:** Recognize the polynomial as a perfect square trinomial. $$4m^2 - 20m + 25 = (2m - 5)^2$$ 3. **Step 2:** The side length is $$2m - 5$$ meters. --- 1. **Problem:** A rectangular park has an area of $$x^2 - 81$$ square meters. Factor completely to find possible dimensions. 2. **Step 1:** Recognize the difference of squares. $$x^2 - 81 = (x)^2 - (9)^2 = (x - 9)(x + 9)$$ 3. **Step 2:** Possible dimensions are $$x - 9$$ meters and $$x + 9$$ meters. --- 1. **Problem:** A rectangular lot has an area of $$x^2 + 9x + 20$$ square meters. Factor completely. 2. **Step 1:** Find two numbers that multiply to $$20$$ and add to $$9$$: $$4$$ and $$5$$. 3. **Step 2:** Factor the polynomial: $$x^2 + 9x + 20 = (x + 4)(x + 5)$$ 4. **Step 3:** Possible dimensions are $$x + 4$$ meters and $$x + 5$$ meters. --- 1. **Problem:** A garden has an area of $$9y^2 - 49$$ square meters. Factor the polynomial. 2. **Step 1:** Recognize the difference of squares. $$9y^2 - 49 = (3y)^2 - 7^2 = (3y - 7)(3y + 7)$$ 3. **Step 2:** Possible dimensions are $$3y - 7$$ meters and $$3y + 7$$ meters. --- **Final answers:** 1. Side length: $$x + 5$$ 2. Side length: $$2m - 5$$ 3. Dimensions: $$(x - 9)$$ and $$(x + 9)$$ 4. Dimensions: $$(x + 4)$$ and $$(x + 5)$$ 5. Dimensions: $$(3y - 7)$$ and $$(3y + 7)$$