Factor Trinomials

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Factor Trinomials · Grade 9 · Worksheet 1

1. A rectangular garden has an area of (6x² + 19x + 10) square meters. If the length of the garden is (3x + 2) meters, what is the width of the garden in terms of x? Answer: (2x + 5) Solution: Area = 6x² + 19x + 10 Length = 3x + 2 Width = ? Recall the formula for the area of a rectangle. Area = Length × Width So, Width = Area ÷ Length Substitute the given expressions.
   Full step-by-step solution

   We are given: Area = 6x² + 19x + 10 Length = 3x + 2 Width = ? Step 1: Recall the formula for the area of a rectangle. Area = Length × Width So, Width = Area ÷ Length Step 2: Substitute the given expressions. Width = (6x² + 19x + 10) ÷ (3x + 2) Step 3: Factor the numerator if possible, since we suspect it is divisible by (3x + 2). We need two numbers that multiply to 6x² × 10 = 60x² and add to 19x. The numbers 15x and 4x work because 15x × 4x = 60x² and 15x + 4x = 19x. Step 4: Rewrite the middle term using 15x and 4x. 6x² + 19x + 10 = 6x² + 15x + 4x + 10 Step 5: Factor by grouping. Group the first two terms: 6x² + 15x = 3x(2x + 5) Group the last two terms: 4x + 10 = 2(2x + 5) Step 6: Factor out the common factor (2x + 5). 3x(2x + 5) + 2(2x + 5) = (2x + 5)(3x + 2) Step 7: Now substitute back into the Width expression. Width = [(2x + 5)(3x + 2)] ÷ (3x + 2) Step 8: Cancel the common factor (3x + 2) (assuming x is such that 3x + 2 ≠ 0). Width = 2x + 5 Final answer: 2x + 5 meters

2. Aroha is constructing a rectangular wooden frame for a school art installation. The area of the frame's surface (in square meters) is given by the quadratic expression 9x² + 39x + 42, where x is a positive integer representing the width in meters. The length and width of the frame are both binomial expressions with integer coefficients. Using the AC method, factor this quadratic expression to find the binomial expressions that represent the possible dimensions of the frame. Answer: (3x + 7)(3x + 6) Solution: Identify coefficients: a = 9, b = 39, c = 42. Step 2: Compute a × c = 9 × 42 = 378. Step 3: Find two numbers that multiply to 378 and add to 39.
   Full step-by-step solution

   Step 1: Identify coefficients: a = 9, b = 39, c = 42. Step 2: Compute a × c = 9 × 42 = 378. Step 3: Find two numbers that multiply to 378 and add to 39. The factor pairs of 378 include 18 and 21 because 18 × 21 = 378 and 18 + 21 = 39. Step 4: Rewrite the middle term 39x as 18x + 21x: 9x² + 18x + 21x + 42. Step 5: Group the terms: (9x² + 18x) + (21x + 42). Step 6: Factor each group: 9x² + 18x = 9x(x + 2) and 21x + 42 = 21(x + 2). Step 7: Factor out the common binomial (x + 2): (9x + 21)(x + 2). Step 8: Simplify the first binomial by factoring out a 3: 9x + 21 = 3(3x + 7). So the factored form becomes 3(3x + 7)(x + 2). However, note that the problem asks for binomial expressions with integer coefficients representing dimensions. Since 3 is a constant factor, the dimensions could be (3x + 7) and (3x + 6) if we consider a different grouping? Let's re-check: Actually, after factoring out the common binomial (x + 2), we have (9x + 21)(x + 2). Factoring 3 from the first binomial gives 3(3x + 7)(x + 2). To express as two binomials with integer coefficients, we can multiply the 3 into one binomial: (3x + 7)(3x + 6) because 3(x + 2) = 3x + 6. So the final factored form is (3x + 7)(3x + 6). Verification: (3x + 7)(3x + 6) = 9x² + 18x + 21x + 42 = 9x² + 39x + 42. The factored form is (3x + 7)(3x + 6).

3. A physics class is designing a catapult that launches projectiles. The height of a projectile above ground is modeled by the quadratic equation h(t) = -16t² + 64t + 80, where h is height in feet and t is time in seconds. Factor this quadratic expression completely to find the time when the projectile hits the ground. Answer: (t - 5)(-16t - 16) Solution: Start with the quadratic: -16t² + 64t + 80 Factor out the greatest common factor, which is -16: -16(t² - 4t - 5) Now factor the quadratic inside the parentheses: t² - 4t - 5 Find two numbers that multiply to -5 and add to -4: -5 and 1 Write the factored form: (t - 5)(t + 1) Include the -16 we…
   Full step-by-step solution

   Step 1: Start with the quadratic: -16t² + 64t + 80 Step 2: Factor out the greatest common factor, which is -16: -16(t² - 4t - 5) Step 3: Now factor the quadratic inside the parentheses: t² - 4t - 5 Step 4: Find two numbers that multiply to -5 and add to -4: -5 and 1 Step 5: Write the factored form: (t - 5)(t + 1) Step 6: Include the -16 we factored out: -16(t - 5)(t + 1) Step 7: The projectile hits the ground when h(t) = 0, so we set the factored form equal to zero: -16(t - 5)(t + 1) = 0 Step 8: The time when the projectile hits the ground is t = 5 seconds (we disregard t = -1 since time cannot be negative). The completely factored form is -16(t - 5)(t + 1).

4. A physics class is designing a catapult that launches projectiles. The height of a projectile above ground level is modeled by the quadratic equation h(t) = -16t² + 64t + 80, where h is the height in feet and t is the time in seconds. To analyze the projectile's flight time, factor this quadratic expression using the AC method to determine when the projectile will return to ground level. Answer: (t - 5)(-16t - 16) Solution: The AC method for factoring quadratics involves finding two numbers that multiply to the product of the leading coefficient and constant term, and add to the middle coefficient. For quadratics with negative leading coefficients, it's often helpful to factor out -1 first.
   Full step-by-step solution

   The AC method for factoring quadratics involves finding two numbers that multiply to the product of the leading coefficient and constant term, and add to the middle coefficient. For quadratics with negative leading coefficients, it's often helpful to factor out -1 first. This technique is useful in physics for analyzing projectile motion and finding key points like maximum height and time of flight.

5. Isabella is creating a geometric design using a rectangular grid. The area of the entire rectangle is given by the quadratic expression 12x² + 40x + 25 square units. She partitions the rectangle into four smaller rectangles using the area model method, where the length and width of the whole rectangle are binomial expressions with integer coefficients. What is the factored form representing the dimensions of Isabella's rectangle? Answer: (2x+5)(6x+5) Solution: Identify a, b, c in 12x² + 40x + 25. Here a = 12, b = 40, c = 25. Multiply a and c: 12 × 25 = 300.
   Full step-by-step solution

   Step 1: Identify a, b, c in 12x² + 40x + 25. Here a = 12, b = 40, c = 25. Step 2: Multiply a and c: 12 × 25 = 300. Step 3: Find two numbers that multiply to 300 and add to 40. The numbers are 10 and 30 because 10 × 30 = 300 and 10 + 30 = 40. Step 4: Rewrite the middle term using these numbers: 12x² + 10x + 30x + 25. Step 5: Factor by grouping. Group the first two terms and the last two terms: (12x² + 10x) + (30x + 25). Step 6: Factor out the greatest common factor from each group: 2x(6x + 5) + 5(6x + 5). Step 7: Factor out the common binomial (6x + 5): (2x + 5)(6x + 5). The factored form is (2x+5)(6x+5).

6. A rectangular garden has an area represented by the expression 2x² + 7x + 6 square meters. The length and width are both binomial expressions with integer coefficients. If the length is longer than the width, what is the expression for the length of the garden? Answer: (2x+3) Solution: Area = 2x² + 7x + 6 Length and width are binomials with integer coefficients. Length > Width. We need to find the expression for the length.
   Full step-by-step solution

   We are given: Area = 2x² + 7x + 6 Length and width are binomials with integer coefficients. Length > Width. We need to find the expression for the length. --- **Step 1: Factor the quadratic expression** We factor 2x² + 7x + 6. Multiply the coefficient of x² (which is 2) by the constant term (6): 2 × 6 = 12. We need two numbers that multiply to 12 and add to 7 (the coefficient of x). Those numbers are 3 and 4 because 3 × 4 = 12 and 3 + 4 = 7. --- **Step 2: Rewrite the middle term using these numbers** 2x² + 7x + 6 = 2x² + 3x + 4x + 6 --- **Step 3: Factor by grouping** Group the terms: (2x² + 3x) + (4x + 6) Factor each group: x(2x + 3) + 2(2x + 3) --- **Step 4: Factor out the common binomial** (2x + 3)(x + 2) So the area factors as: Area = (2x + 3)(x + 2) --- **Step 5: Identify length and width** Length and width are (2x + 3) and (x + 2). Since length > width, compare them: (2x + 3) − (x + 2) = x + 1. For x > 0 (reasonable for a garden's dimensions), x + 1 > 0, so 2x + 3 > x + 2. Thus: Length = 2x + 3 Width = x + 2 --- **Final answer:** Length = 2x + 3

7. Factor: 8x² + 14x - 15 Answer: (4x - 3)(2x + 5) Solution: Identify a = 8, b = 14, c = -15. Multiply a and c: 8 × (-15) = -120. Find two numbers that multiply to -120 and add to 14.
   Full step-by-step solution

   Step 1: Identify a = 8, b = 14, c = -15. Step 2: Multiply a and c: 8 × (-15) = -120. Step 3: Find two numbers that multiply to -120 and add to 14. The numbers are 20 and -6 because 20 × (-6) = -120 and 20 + (-6) = 14. Step 4: Rewrite the middle term using these numbers: 8x² + 20x - 6x - 15. Step 5: Factor by grouping: (8x² + 20x) + (-6x - 15). Step 6: Factor out the greatest common factor from each group: 4x(2x + 5) - 3(2x + 5). Step 7: Factor out the common binomial (2x + 5): (2x + 5)(4x - 3). The answer is (4x - 3)(2x + 5).