TEKS A.10E Worksheets — Algebra I Factor, if possible, trinomials with real factors

Fatima is planting a garden and wants to create a rectangular flower bed. The length of the flower bed is 3 meters more than twice its width. If the area of the flower bed is 40 square meters, which equation can be used to find the width, w, of the flower bed?

1. w(2w + 3) = 40 ✓
2. w(2w - 3) = 40
3. w(w + 3) = 40
4. w(3w + 2) = 40

Why: To find the width of the flower bed, we can express the length in terms of the width: length = 2w + 3. The area of a rectangle is given by length × width. Thus, we can write the equation as w(2w + 3) = 40. This matches the first choice, making it the correct answer.

Mia is planning a fundraiser for her school in McAllen, Texas, where she will sell homemade tamales. The number of tamales she makes can be represented by the expression x² + 7x + 10, where x is the number of batches made. If Mia wants to factor this expression to find the number of tamales based on how many batches she makes, which expression should she use?

1. (x + 2)(x + 5) ✓
2. (x + 1)(x + 10)
3. (x + 7)(x + 3)
4. (x - 2)(x - 5)

Why: To factor the quadratic expression x² + 7x + 10, we need to find two numbers that multiply to 10 (the constant term) and add up to 7 (the coefficient of the x term). The numbers 2 and 5 meet these criteria. Therefore, the factored form is (x + 2)(x + 5), which represents the number of tamales Mia makes based on the batches.

Javier is helping his family run a peach orchard in the Texas Hill Country. They plan to plant a new section of the orchard that can hold a total of 60 trees. The number of trees they can plant is represented by the trinomial x² - 10x + 21. What are the factors of this trinomial that represent the total trees they will plant in this new section?

1. (x - 3)(x - 7) ✓
2. (x - 1)(x - 21)
3. (x + 3)(x - 7)
4. (x - 2)(x - 5)

Why: To factor the trinomial x² - 10x + 21, we need to find two numbers that multiply to 21 (the constant term) and add up to -10 (the coefficient of x). The numbers -3 and -7 meet these conditions. Thus, we can write the trinomial as (x - 3)(x - 7). Therefore, the correct factors are (x - 3)(x - 7).

Andrew wants to create a rectangular garden in his backyard in Fort Worth. He plans to use a length of (2x + 3) feet and a width of (x - 1) feet. What is the area of the garden expressed as a trinomial in standard form? Factor the trinomial if possible.

1. 2x² + x - 3 ✓
2. 2x² + 5x - 3
3. 2x² - x - 3
4. 2x² + 3x - 3

Why: To find the area of the garden, we multiply the length and the width: (2x + 3)(x - 1) = 2x² - 2x + 3x - 3 = 2x² + x - 3. The resulting trinomial is in standard form, and it can be factored as (2x + 3)(x - 1). Therefore, the correct answer is 2x² + x - 3.

Avery is helping organize a local charity event in Fredericksburg, Texas, that involves selling homemade pies. The number of pies sold can be represented by the quadratic equation p² + 7p + 10. Which of the following is the correct factorization of this trinomial?

1. (p + 5)(p + 2) ✓
2. (p + 4)(p + 3)
3. (p + 5)(p + 3)
4. (p + 10)(p + 1)

Why: To factor the trinomial p² + 7p + 10, we need to find two numbers that multiply to 10 (the constant term) and add up to 7 (the coefficient of the middle term p). The numbers 5 and 2 meet these criteria because 5 * 2 = 10 and 5 + 2 = 7. Thus, the correct factorization is (p + 5)(p + 2).

Imani is planning a field trip for her class to the Bluebonnet fields in Texas. The total cost of the trip can be represented by the trinomial expression 2x² + 16x + 24, where x represents the number of students attending. What expression represents the factored form of the total cost of the trip?

1. 2(x + 6)(x + 2) ✓
2. 2(x + 12)(x + 1)
3. 2(x + 8)(x + 3)
4. 2(x + 4)(x + 6)

Why: To factor the trinomial 2x² + 16x + 24, first, we factor out the greatest common factor, which is 2. This gives us 2(x² + 8x + 12). Next, we need to factor the quadratic expression x² + 8x + 12. We find two numbers that multiply to 12 and add up to 8, which are 6 and 2. Thus, we can write it as 2(x + 6)(x + 2). Therefore, the correct answer is 2(x + 6)(x + 2).

While planning a community garden in McAllen, Sanaa created a quadratic expression to represent the area of the garden in square feet. The expression is x² + 8x + 15. What are the correct factors of this expression?

1. (x + 3)(x + 5) ✓
2. (x + 2)(x + 6)
3. (x + 1)(x + 15)
4. (x + 4)(x + 4)

Why: To factor the quadratic expression x² + 8x + 15, we look for two numbers that multiply to 15 and add to 8. The numbers 3 and 5 satisfy both conditions, so we can express the trinomial as (x + 3)(x + 5). Thus, the correct answer is (x + 3)(x + 5).

Riley is studying the growth of pecan trees in Texas. He models the growth of a pecan tree's height with the quadratic expression h(x) = x² + 8x + 16, where h represents the height in feet and x is the number of years since planting. Which expression best represents the factored form of the height of the tree after x years?

1. (x + 4)(x + 4) ✓
2. (x + 2)(x + 8)
3. (x + 1)(x + 16)
4. (x + 4)(x + 2)

Why: To factor the expression h(x) = x² + 8x + 16, we look for two numbers that multiply to 16 (the constant term) and add up to 8 (the coefficient of x). The numbers 4 and 4 meet these criteria. Thus, the factored form is (x + 4)(x + 4) or (x + 4)².