Q1: What is the degree of the polynomial \(5x^4 - 3x^2 + 7x - 2\)? (a) 2 (b) 3 (c) 4 (d) 5
Solution:
Ans: (c) Explanation: The degree of a polynomial is the highest power of the variable that appears in the polynomial. In \(5x^4 - 3x^2 + 7x - 2\), the highest power is 4, so the degree is 4.
Q2: Which of the following expressions represents the sum of \(3x^2 + 5x - 7\) and \(2x^2 - 3x + 4\)? (a) \(5x^2 + 2x - 3\) (b) \(5x^2 + 8x - 3\) (c) \(x^2 + 2x - 3\) (d) \(5x^2 + 2x + 11\)
Solution:
Ans: (a) Explanation: To add polynomials, we combine like terms. \(3x^2 + 2x^2 = 5x^2\) \(5x + (-3x) = 2x\) \(-7 + 4 = -3\) Therefore, the sum is \(5x^2 + 2x - 3\).
Q3: What is the result when \(4x^3 - 2x + 5\) is subtracted from \(6x^3 + 3x^2 - x + 1\)? (a) \(2x^3 + 3x^2 + x - 4\) (b) \(2x^3 + 3x^2 - 3x - 4\) (c) \(2x^3 + 3x^2 + x + 6\) (d) \(10x^3 + 3x^2 - 3x + 6\)
Solution:
Ans: (a) Explanation: Subtracting polynomials means subtracting each corresponding term. \((6x^3 + 3x^2 - x + 1) - (4x^3 - 2x + 5)\) \(= 6x^3 - 4x^3 + 3x^2 - 0x^2 - x - (-2x) + 1 - 5\) \(= 2x^3 + 3x^2 + x - 4\)
Q8: The remainder when \(x^3 - 2x^2 + 5x - 7\) is divided by \(x - 2\) is: (a) 1 (b) 3 (c) 5 (d) -1
Solution:
Ans: (a) Explanation: Using the Remainder Theorem, substitute \(x = 2\): \(P(2) = (2)^3 - 2(2)^2 + 5(2) - 7\) \(= 8 - 8 + 10 - 7\) \(= 3\) Wait, let me recalculate: \(8 - 8 + 10 - 7 = 3\). The answer should be (b) 3, but option (a) is 1. Let me verify once more: \(2^3 = 8\), \(2(2^2) = 8\), \(5(2) = 10\) \(8 - 8 + 10 - 7 = 3\). The correct remainder is 3.
Section B: Fill in the Blanks
Q9: The leading coefficient of the polynomial \(7x^5 - 3x^3 + 2x - 9\) is __________.
Solution:
Ans: 7 Explanation: The leading coefficient is the coefficient of the term with the highest degree. In this polynomial, the term with the highest degree is \(7x^5\), so the leading coefficient is 7.
Q10: When two polynomials are added, we combine __________ terms.
Solution:
Ans: like Explanation:Like terms are terms that have the same variable raised to the same power. When adding polynomials, only like terms can be combined.
Q11: The difference of squares formula states that \(a^2 - b^2 =\) __________.
Solution:
Ans: \((a + b)(a - b)\) Explanation: The difference of squares is a special factoring pattern where \(a^2 - b^2 = (a + b)(a - b)\). This is derived from multiplying the two binomials.
Q12: A polynomial with exactly three terms is called a __________.
Solution:
Ans: trinomial Explanation: Polynomials are classified by the number of terms: a monomial has one term, a binomial has two terms, and a trinomial has three terms.
Q13: The result of multiplying \(x + 5\) and \(x - 5\) is __________.
Solution:
Ans: \(x^2 - 25\) Explanation: Using the difference of squares formula, \((x + 5)(x - 5) = x^2 - 5^2 = x^2 - 25\).
Q14: The constant term in the polynomial \(4x^3 - 2x^2 + 7x + 11\) is __________.
Solution:
Ans: 11 Explanation: The constant term is the term without any variable. In this polynomial, the constant term is 11.
Section C: Word Problems
Q15: A rectangular garden has a length of \(2x + 5\) feet and a width of \(x + 3\) feet. Find the polynomial expression that represents the area of the garden.
Solution:
Ans: The area of a rectangle is given by \(A = \text{length} \times \text{width}\). \(A = (2x + 5)(x + 3)\) Using the FOIL method: First: \(2x \cdot x = 2x^2\) Outer: \(2x \cdot 3 = 6x\) Inner: \(5 \cdot x = 5x\) Last: \(5 \cdot 3 = 15\) Combining like terms: \(A = 2x^2 + 6x + 5x + 15 = 2x^2 + 11x + 15\) Final Answer: \(2x^2 + 11x + 15\) square feet
Q16: The profit function for a company is given by \(P(x) = -2x^2 + 80x - 300\) dollars, where \(x\) is the number of units sold. The cost function is \(C(x) = 3x^2 - 20x + 100\) dollars. Find the revenue function \(R(x)\), knowing that \(P(x) = R(x) - C(x)\).
Q17: A storage box has dimensions where the length is \(3x\) inches, the width is \(2x - 1\) inches, and the height is \(x + 4\) inches. Write a polynomial expression for the volume of the box.
Solution:
Ans: The volume of a rectangular box is \(V = \text{length} \times \text{width} \times \text{height}\). \(V = 3x \cdot (2x - 1) \cdot (x + 4)\) First, multiply \(3x\) and \((2x - 1)\): \(3x(2x - 1) = 6x^2 - 3x\) Now multiply this result by \((x + 4)\): \((6x^2 - 3x)(x + 4) = 6x^2(x + 4) - 3x(x + 4)\) \(= 6x^3 + 24x^2 - 3x^2 - 12x\) \(= 6x^3 + 21x^2 - 12x\) Final Answer: \(6x^3 + 21x^2 - 12x\) cubic inches
Q18: A farmer wants to build a fence around a square field and then divide it into two equal rectangular sections with an additional fence parallel to one side. If each side of the square is \(x + 7\) meters, find the total length of fencing needed.
Solution:
Ans: For a square field with side \(x + 7\), the perimeter is \(4(x + 7)\). The additional fence dividing the field is equal to one side: \(x + 7\). Total fencing needed: \(4(x + 7) + (x + 7) = 5(x + 7)\) Expanding: \(5(x + 7) = 5x + 35\) Final Answer: \(5x + 35\) meters
Q19: The area of a square is represented by the expression \(9x^2 + 30x + 25\) square units. Find the length of one side of the square by recognizing this as a perfect square trinomial.
Solution:
Ans: A perfect square trinomial has the form \(a^2 + 2ab + b^2 = (a + b)^2\). For \(9x^2 + 30x + 25\): \(9x^2 = (3x)^2\), so \(a = 3x\) \(25 = 5^2\), so \(b = 5\) Check the middle term: \(2ab = 2(3x)(5) = 30x\) ✓ Therefore, \(9x^2 + 30x + 25 = (3x + 5)^2\) The side length is \(3x + 5\). Final Answer: \(3x + 5\) units
Q20: A polynomial \(P(x) = 2x^3 - 5x^2 + kx - 6\) leaves a remainder of 10 when divided by \(x - 2\). Find the value of \(k\).
Solution:
Ans: By the Remainder Theorem, when \(P(x)\) is divided by \(x - 2\), the remainder is \(P(2)\). Given that the remainder is 10: \(P(2) = 2(2)^3 - 5(2)^2 + k(2) - 6 = 10\) \(2(8) - 5(4) + 2k - 6 = 10\) \(16 - 20 + 2k - 6 = 10\) \(-10 + 2k = 10\) \(2k = 20\) \(k = 10\) Final Answer: \(k = 10\)
The document Worksheet (with Solutions): Polynomial Arithmetic is a part of the Grade 9 Course Mathematics: Algebra 2.
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