Polynomial Arithmetic - Accuplacer Aaf

Which of the following is the simplified form of $(x^2 - 2x + 4)(x + 2)$ ?
Answer: $x^3 + 8$ This multiplication follows the sum of cubes formula, $(a^2 - ab + b^2)(a + b) = a^3 + b^3$ , where $a=x$ and $b=2$ . The result is $x^3 + 2^3 = x^3 + 8$ . Alternatively, distribute term by term: $x(x^2 - 2x + 4) + 2(x^2 - 2x + 4) = (x^3 - 2x^2 + 4x) + (2x^2 - 4x + 8)$ . The middle terms, $-2x^2$ and $+2x^2$ , and $+4x$ and $-4x$ , cancel out, leaving $x^3 + 8$ .
If $P(x) = 3x^2 - 5x + 2$ and $Q(x) = x^2 + 4x - 1$ , what is the coefficient of $x$ in $P(x) - 2Q(x)$ ?
Answer: -13 First expand $2Q(x) = 2(x^2 + 4x - 1) = 2x^2 + 8x - 2$ . Then $P(x) - 2Q(x) = (3x^2 - 5x + 2) - (2x^2 + 8x - 2) = 3x^2 - 5x + 2 - 2x^2 - 8x + 2 = x^2 - 13x + 4$ . The coefficient of $x$ is $-13$ . Choice B results from forgetting to distribute the negative sign to $8x$ . Choice C results from incorrectly computing $-5 - (-8) = 3$ . Choice D results from getting the correct magnitude but wrong sign.
If $T(x) = x^2 - x - 6$ and $U(x) = 2x - 1$ , what is the coefficient of $x$ in $T(x) \cdot U(x)$ ?
Answer: -11 $(x^2 - x - 6)(2x - 1) = x^2(2x - 1) - x(2x - 1) - 6(2x - 1) = 2x^3 - x^2 - 2x^2 + x - 12x + 6 = 2x^3 - 3x^2 - 11x + 6$ . The coefficient of $x$ is $-11$ . Choice B results from only considering $-x \cdot (-1) - 6 = 1 - 6 = -5$ and missing some terms. Choice C results from sign errors. Choice D results from incorrect combination of the middle terms.
A company's weekly revenue, $R$ , from selling $x$ units of a product is given by the polynomial $R(x) = 50x - 0.1x^2$ . The weekly cost, $C$ , to produce $x$ units is given by $C(x) = 500 + 10x$ . Which polynomial represents the weekly profit, $P(x) = R(x) - C(x)$ ?
Answer: $-0.1x^2 + 40x - 500$ Profit is calculated as Revenue minus Cost, so $P(x) = R(x) - C(x)$ . Substitute the given polynomials: $P(x) = (50x - 0.1x^2) - (500 + 10x)$ . Distribute the negative sign to the cost polynomial: $50x - 0.1x^2 - 500 - 10x$ . Combine like terms and write in standard form (by decreasing power of x): $-0.1x^2 + (50x - 10x) - 500 = -0.1x^2 + 40x - 500$ .
A box has a length of $(x+4)$ , a width of $(x-1)$ , and a height of $(x)$ . Which polynomial represents the volume of the box?
Answer: $x^3 + 3x^2 - 4x$ The volume of a box is length × width × height. So, the volume is $(x+4)(x-1)(x)$ . First, multiply the two binomials: $(x+4)(x-1) = x^2 - x + 4x - 4 = x^2 + 3x - 4$ . Then, multiply this result by the height, $x$ : $x(x^2 + 3x - 4) = x^3 + 3x^2 - 4x$ .
When $(3x^2 - x + 2)^2$ is expanded, what is the coefficient of $x^3$ ?
Answer: -6 When you encounter a problem asking for a specific coefficient after expanding a squared expression, you need to use the binomial expansion formula or systematically multiply the terms. To find the coefficient of $x^3$ in $(3x^2 - x + 2)^2$ , you can use the fact that when squaring a trinomial, each term in the first factor multiplies with each term in the second factor. The $x^3$ terms come from multiplying terms whose exponents sum to 3. There are two ways to get $x^3$ : - $(3x^2) \cdot (-x) = -3x^3$ - $(-x) \cdot (3x^2) = -3x^3$ Adding these contributions: $-3x^3 + (-3x^3) = -6x^3$ Therefore, the coefficient of $x^3$ is $-6$ . Looking at the wrong answers: Choice A (12) likely comes from forgetting the negative signs and possibly making arithmetic errors. Choice B (6) represents the correct magnitude but with the wrong sign - you might get this if you forgot that both contributing terms are negative. Choice D (-12) suggests doubling errors or miscalculating the products of coefficients. For polynomial expansion problems on the Accuplacer, always track your signs carefully and remember that when finding a specific term's coefficient, you need to identify all the ways that term can be formed from the multiplication. Writing out the key products systematically helps avoid sign errors and missed terms.
Which of the following is equivalent to $(x - 3)(x + 3)(2x - 1)$ ?
Answer: $2x^3 - x^2 - 18x + 9$ First, multiply the first two binomials, which are a difference of squares: $(x - 3)(x + 3) = x^2 - 9$ . Then, multiply this result by the third binomial: $(x^2 - 9)(2x - 1)$ . Using distribution (FOIL): $(x^2)(2x) + (x^2)(-1) + (-9)(2x) + (-9)(-1) = 2x^3 - x^2 - 18x + 9$ .
The perimeter of a triangle is given by the polynomial $15x^2 - 8x + 12$ . Two of the sides have lengths $4x^2 + x - 3$ and $7x^2 - 5x + 9$ . What is the length of the third side?
Answer: $4x^2 - 4x + 6$ First, find the sum of the two known sides: $(4x^2 + x - 3) + (7x^2 - 5x + 9) = 11x^2 - 4x + 6$ . To find the third side, subtract this sum from the perimeter: $(15x^2 - 8x + 12) - (11x^2 - 4x + 6)$ . Distribute the negative: $15x^2 - 8x + 12 - 11x^2 + 4x - 6$ . Combine like terms: $(15x^2 - 11x^2) + (-8x + 4x) + (12 - 6) = 4x^2 - 4x + 6$ .
What is the simplified form of the expression $4x^2 - [x(3x - 1) - (2x^2 - 5x)]$ ?
Answer: $3x^2 - 4x$ First, simplify the expression inside the brackets. Distribute $x$ : $3x^2 - x$ . Distribute the negative: $-(2x^2 - 5x) = -2x^2 + 5x$ . Combine these results inside the bracket: $[3x^2 - x - 2x^2 + 5x] = [x^2 + 4x]$ . Now, substitute this back into the original expression: $4x^2 - [x^2 + 4x]$ . Distribute the outer negative sign: $4x^2 - x^2 - 4x$ . Combine like terms to get $3x^2 - 4x$ .
Which of the following is equivalent to the expression $(2x + 1)^2 - (2x - 1)^2$ ?
Answer: $8x$ First, expand each squared binomial. $(2x + 1)^2 = (2x)^2 + 2(2x)(1) + 1^2 = 4x^2 + 4x + 1$ . $(2x - 1)^2 = (2x)^2 - 2(2x)(1) + 1^2 = 4x^2 - 4x + 1$ . Now, subtract the second result from the first: $(4x^2 + 4x + 1) - (4x^2 - 4x + 1)$ . Distribute the negative: $4x^2 + 4x + 1 - 4x^2 + 4x - 1$ . The $4x^2$ terms cancel and the $1$ terms cancel, leaving $4x + 4x = 8x$ .
The sum of two polynomials is $6x^2 - 10x + 5$ . If one of the polynomials is $-2x^2 + 3x - 7$ , what is the other polynomial?
Answer: $8x^2 - 13x + 12$ To find the other polynomial, subtract the known polynomial from the sum: $(6x^2 - 10x + 5) - (-2x^2 + 3x - 7)$ . Distribute the negative sign to the second polynomial: $6x^2 - 10x + 5 + 2x^2 - 3x + 7$ . Combine like terms: $(6x^2 + 2x^2) + (-10x - 3x) + (5 + 7) = 8x^2 - 13x + 12$ .
Which of the following is the result of subtracting $(2x^2 - 7x + 3)$ from $(-5x^2 + 4x - 8)$ ?
Answer: $-7x^2 + 11x - 11$ To subtract $(2x^2 - 7x + 3)$ from $(-5x^2 + 4x - 8)$ , set up the expression: $(-5x^2 + 4x - 8) - (2x^2 - 7x + 3)$ . Distribute the negative sign to each term in the second polynomial: $-5x^2 + 4x - 8 - 2x^2 + 7x - 3$ . Combine like terms: $(-5x^2 - 2x^2) + (4x + 7x) + (-8 - 3) = -7x^2 + 11x - 11$ .
The expression $(ax^2 - 5x + 2) - (3x^2 + bx - 4)$ is simplified to $-x^2 + 8x + c$ . What is the value of $a - b$ ?
Answer: $15$ First, simplify the given expression: $ax^2 - 5x + 2 - 3x^2 - bx + 4 = (a-3)x^2 + (-5-b)x + 6$ . Now, equate the coefficients of this simplified form with $-x^2 + 8x + c$ . For the $x^2$ term: $a - 3 = -1$ , which means $a = 2$ . For the $x$ term: $-5 - b = 8$ , which means $-b = 13$ or $b = -13$ . The question asks for $a - b$ . Substitute the values: $2 - (-13) = 2 + 13 = 15$ .
When $(2x - 3)(x^2 + 5x - 4)$ is expanded and simplified, what is the coefficient of $x^2$ ?
Answer: 7 Using the distributive property: $(2x - 3)(x^2 + 5x - 4) = 2x(x^2 + 5x - 4) - 3(x^2 + 5x - 4) = 2x^3 + 10x^2 - 8x - 3x^2 - 15x + 12 = 2x^3 + 7x^2 - 23x + 12$ . The coefficient of $x^2$ is $7$ . Choice A is the coefficient of $-3x^2$ alone. Choice C is the coefficient of $10x^2$ alone. Choice D is the coefficient of $x$ .
Which expression is the result of the multiplication $(7a^3 - 4b^2)(7a^3 + 4b^2)$ ?
Answer: $49a^6 - 16b^4$ This is a product of the sum and difference of two terms, which follows the pattern $(x - y)(x + y) = x^2 - y^2$ . Here, $x = 7a^3$ and $y = 4b^2$ . So, the result is $(7a^3)^2 - (4b^2)^2$ . Squaring each part gives $49a^{(3*2)} - 16b^{(2*2)} = 49a^6 - 16b^4$ .
If $A(x) = 2x^2 - 3x + 1$ , $B(x) = x^2 + 5x - 2$ , and $C(x) = 3x - 4$ , what is the coefficient of $x^2$ in $A(x) - B(x) + C(x)$ ?
Answer: 1 $A(x) - B(x) + C(x) = (2x^2 - 3x + 1) - (x^2 + 5x - 2) + (3x - 4) = 2x^2 - 3x + 1 - x^2 - 5x + 2 + 3x - 4 = (2-1)x^2 + (-3-5+3)x + (1+2-4) = x^2 - 5x - 1$ . The coefficient of $x^2$ is 1. Choice A results from incorrectly computing $2-1-1=0$ . Choice C results from only considering the coefficient 2 from $A(x)$ and forgetting to subtract $B(x)$ . Choice D results from adding all $x^2$ coefficients: $2+1+3=6$ .
A rectangular garden has a length of $(3x + 5)$ meters and a width of $(2x - 1)$ meters. Inside the garden, there is a square fountain with a side length of $(x - 2)$ meters. Which polynomial represents the area of the garden not covered by the fountain?
Answer: $5x^2 + 11x - 9$ The area of the garden is $(3x + 5)(2x - 1) = 6x^2 - 3x + 10x - 5 = 6x^2 + 7x - 5$ . The area of the fountain is $(x - 2)^2 = x^2 - 4x + 4$ . To find the remaining area, subtract the fountain's area from the garden's area: $(6x^2 + 7x - 5) - (x^2 - 4x + 4) = 6x^2 + 7x - 5 - x^2 + 4x - 4 = 5x^2 + 11x - 9$ .
Which expression results from simplifying $(3a^2 - 2ab + 5b^2) - (a^2 - 6ab - 3b^2)$ ?
Answer: $2a^2 + 4ab + 8b^2$ To subtract the polynomials, distribute the negative sign to each term in the second polynomial: $3a^2 - 2ab + 5b^2 - a^2 + 6ab + 3b^2$ . Then, group and combine like terms: $(3a^2 - a^2) + (-2ab + 6ab) + (5b^2 + 3b^2)$ . This simplifies to $2a^2 + 4ab + 8b^2$ .
What is the degree of the polynomial that results from $(x^3 - 2x + 1)(x^2 + 3x - 4) + (x^4 - 2x^5)$ ?
Answer: 5 The product $(x^3 - 2x + 1)(x^2 + 3x - 4)$ has highest degree term $x^3 \cdot x^2 = x^5$ with coefficient 1. When we add $(x^4 - 2x^5)$ , the $x^5$ terms combine as $1x^5 + (-2x^5) = -x^5$ . Since this doesn't equal zero, the degree remains 5. Choice A incorrectly assumes the $x^5$ terms cancel completely. Choice C incorrectly adds the degrees $3 + 2 + 1 = 6$ . Choice D incorrectly multiplies the degrees.
What is the coefficient of the $x^2$ term in the simplified expansion of $(2x^2 - 5x + 3)(x - 4)$ ?
Answer: $-13$ To find the $x^2$ term, we only need to find the products of terms that result in $x^2$ . These are: (the $x^2$ term from the first polynomial) × (the constant term from the second) AND (the $x$ term from the first) × (the $x$ term from the second). First product: $(2x^2)(-4) = -8x^2$ . Second product: $(-5x)(x) = -5x^2$ . Add these results: $-8x^2 + (-5x^2) = -13x^2$ . The coefficient is -13.

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