12 multiple-choice questions, progressively harder.
If P=5x2−2x+1P = 5x^2 - 2x + 1P=5x2−2x+1 and Q=2x2+3x−4Q = 2x^2 + 3x - 4Q=2x2+3x−4, what is P−QP - QP−Q?
Solution
Correct answer: B
Compute P−QP - QP−Q by distributing the minus sign across QQQ.
5x2−2x+1−2x2−3x+4=3x2−5x+55x^2 - 2x + 1 - 2x^2 - 3x + 4 = 3x^2 - 5x + 55x2−2x+1−2x2−3x+4=3x2−5x+5
Each term of QQQ flips: +3x+3x+3x to −3x-3x−3x and −4-4−4 to +4+4+4.
If P=2x3−x+4P = 2x^3 - x + 4P=2x3−x+4 and Q=x3+2x2−3Q = x^3 + 2x^2 - 3Q=x3+2x2−3, what is Q−PQ - PQ−P?
Compute Q−PQ - PQ−P by distributing the minus sign across PPP.
x3+2x2−3−2x3+x−4=−x3+2x2+x−7x^3 + 2x^2 - 3 - 2x^3 + x - 4 = -x^3 + 2x^2 + x - 7x3+2x2−3−2x3+x−4=−x3+2x2+x−7
The −x-x−x in PPP flips to +x+x+x, and the +4+4+4 flips to −4-4−4.
What must the constant ccc be so that (x2+3x+c)+(2x2−x−5)(x^2 + 3x + c) + (2x^2 - x - 5)(x2+3x+c)+(2x2−x−5) has no constant term?
Correct answer: C
The constant term of the sum is c+(−5)c + (-5)c+(−5). Set it equal to 000.
c−5=0 ⇒ c=5c - 5 = 0 \;\Rightarrow\; c = 5c−5=0⇒c=5
For which value of ccc does the difference (x2+cx+3)−(x2−4x+c)(x^2 + cx + 3) - (x^2 - 4x + c)(x2+cx+3)−(x2−4x+c) have no xxx term?
Subtract; the x2x^2x2 terms cancel, leaving (c+4)x+(3−c)(c + 4)x + (3 - c)(c+4)x+(3−c). No xxx term means its coefficient is 000.
c+4=0 ⇒ c=−4c + 4 = 0 \;\Rightarrow\; c = -4c+4=0⇒c=−4
Simplify: x2−(3x2−(2x2+x))x^2 - (3x^2 - (2x^2 + x))x2−(3x2−(2x2+x)).
Correct answer: A
Work from the inside out. First the inner subtraction.
3x2−(2x2+x)=3x2−2x2−x=x2−x3x^2 - (2x^2 + x) = 3x^2 - 2x^2 - x = x^2 - x3x2−(2x2+x)=3x2−2x2−x=x2−x
Then the outer subtraction.
x2−(x2−x)=x2−x2+x=xx^2 - (x^2 - x) = x^2 - x^2 + x = xx2−(x2−x)=x2−x2+x=x
The polynomial MMM satisfies M−(x2−2x+5)=3x2+x−2M - (x^2 - 2x + 5) = 3x^2 + x - 2M−(x2−2x+5)=3x2+x−2. What is MMM?
Solve for MMM by adding (x2−2x+5)(x^2 - 2x + 5)(x2−2x+5) to both sides.
M=(3x2+x−2)+(x2−2x+5)=4x2−x+3M = (3x^2 + x - 2) + (x^2 - 2x + 5) = 4x^2 - x + 3M=(3x2+x−2)+(x2−2x+5)=4x2−x+3
The sum of two polynomials is 6x2−x+26x^2 - x + 26x2−x+2. One of them is 4x2+3x−54x^2 + 3x - 54x2+3x−5. What is the other?
Correct answer: D
The other polynomial is the sum minus the known one.
(6x2−x+2)−(4x2+3x−5)=2x2−4x+7(6x^2 - x + 2) - (4x^2 + 3x - 5) = 2x^2 - 4x + 7(6x2−x+2)−(4x2+3x−5)=2x2−4x+7
Each term of the known polynomial flips: +3x+3x+3x to −3x-3x−3x and −5-5−5 to +5+5+5.
For what values of aaa and bbb is (ax2+4x+b)−(2x2+4x−3)(ax^2 + 4x + b) - (2x^2 + 4x - 3)(ax2+4x+b)−(2x2+4x−3) equal to x2+7x^2 + 7x2+7?
Subtract; the 4x4x4x terms cancel, leaving (a−2)x2+(b+3)(a - 2)x^2 + (b + 3)(a−2)x2+(b+3). Match this to x2+7x^2 + 7x2+7.
a−2=1 ⇒ a=3a - 2 = 1 \;\Rightarrow\; a = 3a−2=1⇒a=3
b+3=7 ⇒ b=4b + 3 = 7 \;\Rightarrow\; b = 4b+3=7⇒b=4
Simplify: (x2−3x)+(2x−5)−(x2−8)(x^2 - 3x) + (2x - 5) - (x^2 - 8)(x2−3x)+(2x−5)−(x2−8).
Remove the parentheses, distributing the last minus sign.
x2−3x+2x−5−x2+8x^2 - 3x + 2x - 5 - x^2 + 8x2−3x+2x−5−x2+8
Combine like terms; the x2x^2x2 terms cancel.
(−3x+2x)+(−5+8)=−x+3(-3x + 2x) + (-5 + 8) = -x + 3(−3x+2x)+(−5+8)=−x+3
Simplify: (x2+x+1)−(x2−x+1)+(x2−x−1)(x^2 + x + 1) - (x^2 - x + 1) + (x^2 - x - 1)(x2+x+1)−(x2−x+1)+(x2−x−1).
Distribute the middle minus sign, then combine each power.
x2+x+1−x2+x−1+x2−x−1x^2 + x + 1 - x^2 + x - 1 + x^2 - x - 1x2+x+1−x2+x−1+x2−x−1
=x2+x−1= x^2 + x - 1=x2+x−1
What is the degree of (x5−x3+2)+(x3−x5+x2)(x^5 - x^3 + 2) + (x^3 - x^5 + x^2)(x5−x3+2)+(x3−x5+x2)?
The x5x^5x5 terms cancel and the x3x^3x3 terms cancel.
x5−x5−x3+x3+x2+2=x2+2x^5 - x^5 - x^3 + x^3 + x^2 + 2 = x^2 + 2x5−x5−x3+x3+x2+2=x2+2
The highest power left is 222, so the degree is 222.
If A−B=4x2−2x+6A - B = 4x^2 - 2x + 6A−B=4x2−2x+6 and B=x2+3x−2B = x^2 + 3x - 2B=x2+3x−2, what is AAA?
Since A−BA - BA−B is known, recover AAA by adding BBB back.
A=(4x2−2x+6)+(x2+3x−2)=5x2+x+4A = (4x^2 - 2x + 6) + (x^2 + 3x - 2) = 5x^2 + x + 4A=(4x2−2x+6)+(x2+3x−2)=5x2+x+4
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