Ans: (a) Explanation: To factor \(x^2 + 7x + 12\), we need two numbers that multiply to 12 and add to 7. These numbers are 3 and 4. Therefore, \(x^2 + 7x + 12 = (x + 3)(x + 4)\). Option (b) gives a sum of 8, option (c) gives a sum of 13, and option (d) would give a positive quadratic with negative middle term.
Q2: What is the greatest common factor (GCF) of \(12x^3y^2 + 18x^2y^3 - 6xy^2\)? (a) \(6xy\) (b) \(6xy^2\) (c) \(3xy^2\) (d) \(12x^2y^2\)
Solution:
Ans: (b) Explanation: The GCF of the coefficients 12, 18, and 6 is 6. For the variables, the lowest power of \(x\) present in all terms is \(x^1\), and the lowest power of \(y\) is \(y^2\). Therefore, the GCF is \(6xy^2\).
Q3: Which of the following is the factored form of \(x^2 - 49\)? (a) \((x - 7)^2\) (b) \((x + 7)^2\) (c) \((x - 7)(x + 7)\) (d) \((x - 49)(x + 1)\)
Solution:
Ans: (c) Explanation: The expression \(x^2 - 49\) is a difference of squares since \(49 = 7^2\). Using the pattern \(a^2 - b^2 = (a - b)(a + b)\), we get \(x^2 - 49 = (x - 7)(x + 7)\). Options (a) and (b) are perfect square trinomials, and option (d) does not multiply correctly.
Ans: (a) Explanation: For factoring trinomials with leading coefficient 2, we need two numbers that multiply to \(2 \times 5 = 10\) and add to 11. These are 10 and 1. Rewriting: \(2x^2 + 10x + x + 5 = 2x(x + 5) + 1(x + 5) = (2x + 1)(x + 5)\). Note that options (a), (b), and (c) appear similar, but (b) and (c) give \(2x^2 + 7x + 5\), which is incorrect.
Q5: Which polynomial is a perfect square trinomial? (a) \(x^2 + 6x + 9\) (b) \(x^2 + 6x + 8\) (c) \(x^2 + 9\) (d) \(x^2 - 6x + 8\)
Solution:
Ans: (a) Explanation: A perfect square trinomial has the form \(a^2 + 2ab + b^2 = (a + b)^2\). For \(x^2 + 6x + 9\), we have \(a = x\), \(b = 3\), and \(2ab = 2(x)(3) = 6x\). Therefore, \(x^2 + 6x + 9 = (x + 3)^2\). The other options do not fit this pattern.
Ans: (b) Explanation: Group the terms: \((6x^3 - 9x^2) + (4x - 6)\). Factor each group: \(3x^2(2x - 3) + 2(2x - 3)\). Factor out the common binomial: \((2x - 3)(3x^2 + 2)\). Options (a) and (b) are equivalent, both are correct forms.
Q7: What is the complete factorization of \(x^3 - 8\)? (a) \((x - 2)(x^2 + 2x + 4)\) (b) \((x - 2)(x^2 - 2x + 4)\) (c) \((x + 2)(x^2 - 2x + 4)\) (d) \((x - 2)^3\)
Solution:
Ans: (a) Explanation: The expression \(x^3 - 8\) is a difference of cubes since \(8 = 2^3\). Using the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), we get \(x^3 - 8 = (x - 2)(x^2 + 2x + 4)\). Option (b) uses the wrong sign for the middle term.
Ans: (a) Explanation: First, factor out the GCF of 5: \(5(x^2 - 9)\). Then recognize \(x^2 - 9\) as a difference of squares: \(x^2 - 9 = (x - 3)(x + 3)\). The complete factorization is \(5(x - 3)(x + 3)\). Option (c) is not fully factored.
Section B: Fill in the Blanks
Q9: The process of writing a polynomial as a product of its factors is called __________.
Solution:
Ans: factoring Explanation:Factoring is the fundamental process in algebra where we express a polynomial as a product of simpler polynomials or monomials.
Q10: The factored form of \(a^2 - b^2\) is __________.
Solution:
Ans: \((a - b)(a + b)\) Explanation: This is the difference of squares pattern, one of the most important factoring formulas in Algebra 2.
Q11: When factoring \(x^2 + bx + c\), we look for two numbers that multiply to __________ and add to __________.
Solution:
Ans: \(c\) and \(b\) Explanation: This is the fundamental principle for factoring trinomials of the form \(x^2 + bx + c\). The two numbers must have a product equal to the constant term and a sum equal to the coefficient of the linear term.
Q12: The formula for factoring the sum of cubes \(a^3 + b^3\) is __________.
Solution:
Ans: \((a + b)(a^2 - ab + b^2)\) Explanation: The sum of cubes formula is essential for factoring cubic polynomials. Note the sign pattern: the binomial has a plus, and the trinomial has minus for the middle term.
Q13: A trinomial of the form \(a^2 + 2ab + b^2\) is called a __________ and factors as __________.
Solution:
Ans: perfect square trinomial and \((a + b)^2\) Explanation: A perfect square trinomial results from squaring a binomial. Recognizing this pattern allows for quick factorization.
Q14: The first step in factoring any polynomial should be to look for the __________.
Solution:
Ans: greatest common factor (GCF) Explanation: Always begin factoring by identifying and extracting the greatest common factor. This simplifies the remaining polynomial and makes further factoring easier.
Section C: Word Problems
Q15: The area of a rectangular garden is represented by the polynomial \(x^2 + 9x + 20\) square feet. Find the dimensions of the garden by factoring the polynomial.
Solution:
Ans: To find the dimensions, we factor \(x^2 + 9x + 20\). We need two numbers that multiply to 20 and add to 9. These numbers are 4 and 5. Therefore: \(x^2 + 9x + 20 = (x + 4)(x + 5)\) The dimensions are length = \((x + 5)\) feet and width = \((x + 4)\) feet. Final Answer: Length = \((x + 5)\) feet, Width = \((x + 4)\) feet
Q16: A square patio has an area of \(4x^2 + 12x + 9\) square meters. What is the length of one side of the patio? (Hint: Factor the expression.)
Solution:
Ans: Since the patio is square, we need to factor \(4x^2 + 12x + 9\). This is a perfect square trinomial: \(4x^2 + 12x + 9 = (2x)^2 + 2(2x)(3) + 3^2\) Using the pattern \(a^2 + 2ab + b^2 = (a + b)^2\): \(4x^2 + 12x + 9 = (2x + 3)^2\) Since Area = (side)2, the side length is \(2x + 3\). Final Answer: Side length = \((2x + 3)\) meters
Q17: The volume of a rectangular box is given by \(6x^3 + 15x^2 + 9x\) cubic inches. If the height of the box is \(3x\) inches, find the area of the base by factoring.
Solution:
Ans: Volume = Base Area × Height Given Volume = \(6x^3 + 15x^2 + 9x\) and Height = \(3x\) First, factor out the GCF from the volume: \(6x^3 + 15x^2 + 9x = 3x(2x^2 + 5x + 3)\) Factor the trinomial: \(2x^2 + 5x + 3 = (2x + 3)(x + 1)\) So Volume = \(3x(2x + 3)(x + 1)\) Since Height = \(3x\), Base Area = \((2x + 3)(x + 1)\) Expanding: Base Area = \(2x^2 + 5x + 3\) Final Answer: Base area = \(2x^2 + 5x + 3\) square inches or \((2x + 3)(x + 1)\) square inches
Q18: A company's profit in dollars is modeled by \(P(x) = -x^2 + 100\), where \(x\) represents hundreds of units sold. Factor this expression to find the values of \(x\) where the profit is zero (break-even points).
Solution:
Ans: Set the profit equal to zero: \(-x^2 + 100 = 0\) Rearrange: \(100 - x^2 = 0\) This is a difference of squares: \(100 - x^2 = (10 - x)(10 + x)\) Setting each factor to zero: \(10 - x = 0\) or \(10 + x = 0\) Solving: \(x = 10\) or \(x = -10\) Since \(x\) represents hundreds of units sold, only positive values make sense. Final Answer: The break-even point is at \(x = 10\) (1000 units sold)
Q19: The difference between the square of a number and 16 times the number can be represented as \(x^2 - 16x\). Factor this expression completely and find the values of \(x\) that make the expression equal to zero.
Solution:
Ans: Factor \(x^2 - 16x\): First, factor out the GCF, which is \(x\): \(x^2 - 16x = x(x - 16)\) To find when the expression equals zero, set \(x(x - 16) = 0\) By the zero product property: \(x = 0\) or \(x - 16 = 0\) Solving: \(x = 0\) or \(x = 16\) Final Answer: Factored form: \(x(x - 16)\); Solutions: \(x = 0\) or \(x = 16\)
Q20: A projectile's height in feet above the ground is given by \(h(t) = -16t^2 + 64t\), where \(t\) is time in seconds. Factor this expression and determine when the projectile hits the ground (height = 0).
Solution:
Ans: Set the height equal to zero: \(-16t^2 + 64t = 0\) Factor out the GCF, which is \(-16t\): \(-16t(t - 4) = 0\) By the zero product property: \(-16t = 0\) or \(t - 4 = 0\) Solving: \(t = 0\) or \(t = 4\) \(t = 0\) is when the projectile is launched. \(t = 4\) is when the projectile hits the ground. Final Answer: The projectile hits the ground at \(t = 4\) seconds
The document Worksheet (with Solutions): Polynomial Factorization is a part of the Grade 9 Course Mathematics: Algebra 2.
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