12 multiple-choice questions, progressively harder.
Factor x3−3x2−2x+6x^3 - 3x^2 - 2x + 6x3−3x2−2x+6 by grouping.
Solution
Correct answer: A
The second pair leads with a negative, so factor out −2-2−2 to match x−3x - 3x−3.
(x3−3x2)+(−2x+6)=x2(x−3)−2(x−3)=(x−3)(x2−2)(x^3 - 3x^2) + (-2x + 6) = x^2(x - 3) - 2(x - 3) = (x - 3)(x^2 - 2)(x3−3x2)+(−2x+6)=x2(x−3)−2(x−3)=(x−3)(x2−2)
Factoring +2+2+2 instead would give 2(3−x)2(3 - x)2(3−x), the wrong binomial.
Factor x3+2x2−5x−10x^3 + 2x^2 - 5x - 10x3+2x2−5x−10 by grouping.
Factor x2x^2x2 from the first pair and −5-5−5 from the second so both give x+2x + 2x+2.
(x3+2x2)+(−5x−10)=x2(x+2)−5(x+2)=(x+2)(x2−5)(x^3 + 2x^2) + (-5x - 10) = x^2(x + 2) - 5(x + 2) = (x + 2)(x^2 - 5)(x3+2x2)+(−5x−10)=x2(x+2)−5(x+2)=(x+2)(x2−5)
The second pair is −5x−10=−5(x+2)-5x - 10 = -5(x + 2)−5x−10=−5(x+2).
Factor 2x2+7x+62x^2 + 7x + 62x2+7x+6 by the AC method.
Correct answer: C
Here ac=2⋅6=12ac = 2 \cdot 6 = 12ac=2⋅6=12 and the sum is 777, met by 333 and 444. Split 7x=3x+4x7x = 3x + 4x7x=3x+4x.
2x2+3x+4x+6=x(2x+3)+2(2x+3)=(2x+3)(x+2)2x^2 + 3x + 4x + 6 = x(2x + 3) + 2(2x + 3) = (2x + 3)(x + 2)2x2+3x+4x+6=x(2x+3)+2(2x+3)=(2x+3)(x+2)
Expanding gives 2x2+7x+62x^2 + 7x + 62x2+7x+6.
Factor x3−5x2+3x−15x^3 - 5x^2 + 3x - 15x3−5x2+3x−15 by grouping.
Correct answer: D
The first pair gives x2(x−5)x^2(x - 5)x2(x−5), and the second pair 3x−15=3(x−5)3x - 15 = 3(x - 5)3x−15=3(x−5) already matches.
(x3−5x2)+(3x−15)=x2(x−5)+3(x−5)=(x−5)(x2+3)(x^3 - 5x^2) + (3x - 15) = x^2(x - 5) + 3(x - 5) = (x - 5)(x^2 + 3)(x3−5x2)+(3x−15)=x2(x−5)+3(x−5)=(x−5)(x2+3)
Both pairs share x−5x - 5x−5.
The polynomial x3+4x2−3x−12x^3 + 4x^2 - 3x - 12x3+4x2−3x−12 factors by grouping. What binomial is common to both pairs?
Factor each pair: the second needs −3-3−3 pulled out to match the first.
x2(x+4)−3(x+4)=(x+4)(x2−3)x^2(x + 4) - 3(x + 4) = (x + 4)(x^2 - 3)x2(x+4)−3(x+4)=(x+4)(x2−3)
The shared binomial is x+4x + 4x+4; x2−3x^2 - 3x2−3 is the leftover factor.
For 3x2−5x−23x^2 - 5x - 23x2−5x−2, the product is ac=−6ac = -6ac=−6 and the sum is −5-5−5. Which numbers split the middle term?
Correct answer: B
The pieces must multiply to −6-6−6 and add to −5-5−5.
(−6)(1)=−6,−6+1=−5(-6)(1) = -6, \qquad -6 + 1 = -5(−6)(1)=−6,−6+1=−5
The others fail one test: −3-3−3 and −2-2−2 multiply to +6+6+6, and −2,3-2, 3−2,3 or −1,6-1, 6−1,6 do not sum to −5-5−5.
Reorder and factor 2x3−5x+6x2−152x^3 - 5x + 6x^2 - 152x3−5x+6x2−15.
Put the terms in descending order first: 2x3+6x2−5x−152x^3 + 6x^2 - 5x - 152x3+6x2−5x−15. Then group.
(2x3+6x2)+(−5x−15)=2x2(x+3)−5(x+3)=(x+3)(2x2−5)(2x^3 + 6x^2) + (-5x - 15) = 2x^2(x + 3) - 5(x + 3) = (x + 3)(2x^2 - 5)(2x3+6x2)+(−5x−15)=2x2(x+3)−5(x+3)=(x+3)(2x2−5)
The second pair is −5x−15=−5(x+3)-5x - 15 = -5(x + 3)−5x−15=−5(x+3).
Factor 4x2+8x+3x+64x^2 + 8x + 3x + 64x2+8x+3x+6 (already split) by grouping.
Factor 4x4x4x from the first pair and 333 from the second pair.
(4x2+8x)+(3x+6)=4x(x+2)+3(x+2)=(x+2)(4x+3)(4x^2 + 8x) + (3x + 6) = 4x(x + 2) + 3(x + 2) = (x + 2)(4x + 3)(4x2+8x)+(3x+6)=4x(x+2)+3(x+2)=(x+2)(4x+3)
Both pairs share x+2x + 2x+2.
Factor 2x2−5x−32x^2 - 5x - 32x2−5x−3 by the AC method.
Here ac=2⋅(−3)=−6ac = 2 \cdot (-3) = -6ac=2⋅(−3)=−6 and the sum is −5-5−5, met by −6-6−6 and 111. Split −5x=−6x+x-5x = -6x + x−5x=−6x+x.
2x2−6x+x−3=2x(x−3)+1(x−3)=(x−3)(2x+1)2x^2 - 6x + x - 3 = 2x(x - 3) + 1(x - 3) = (x - 3)(2x + 1)2x2−6x+x−3=2x(x−3)+1(x−3)=(x−3)(2x+1)
Expanding gives 2x2−5x−32x^2 - 5x - 32x2−5x−3.
The second pair of x3−4x2−5x+20x^3 - 4x^2 - 5x + 20x3−4x2−5x+20 is −5x+20-5x + 20−5x+20. Factoring it so the binomial matches x−4x - 4x−4 gives:
To match the first pair's x−4x - 4x−4, pull out −5-5−5, which turns −5x+20-5x + 20−5x+20 into −5-5−5 times x−4x - 4x−4.
−5x+20=−5(x−4)-5x + 20 = -5(x - 4)−5x+20=−5(x−4)
Then x2(x−4)−5(x−4)=(x−4)(x2−5)x^2(x - 4) - 5(x - 4) = (x - 4)(x^2 - 5)x2(x−4)−5(x−4)=(x−4)(x2−5).
Factor 2x2−7x+62x^2 - 7x + 62x2−7x+6 by the AC method.
Here ac=2⋅6=12ac = 2 \cdot 6 = 12ac=2⋅6=12 and the sum is −7-7−7, met by −3-3−3 and −4-4−4. Split −7x=−3x−4x-7x = -3x - 4x−7x=−3x−4x.
2x2−3x−4x+6=x(2x−3)−2(2x−3)=(2x−3)(x−2)2x^2 - 3x - 4x + 6 = x(2x - 3) - 2(2x - 3) = (2x - 3)(x - 2)2x2−3x−4x+6=x(2x−3)−2(2x−3)=(2x−3)(x−2)
Expanding gives 2x2−7x+62x^2 - 7x + 62x2−7x+6.
Factor 4x2−4x−34x^2 - 4x - 34x2−4x−3 by the AC method.
Here ac=4⋅(−3)=−12ac = 4 \cdot (-3) = -12ac=4⋅(−3)=−12 and the sum is −4-4−4, met by −6-6−6 and 222. Split −4x=−6x+2x-4x = -6x + 2x−4x=−6x+2x.
4x2−6x+2x−3=2x(2x−3)+1(2x−3)=(2x−3)(2x+1)4x^2 - 6x + 2x - 3 = 2x(2x - 3) + 1(2x - 3) = (2x - 3)(2x + 1)4x2−6x+2x−3=2x(2x−3)+1(2x−3)=(2x−3)(2x+1)
Expanding gives 4x2−4x−34x^2 - 4x - 34x2−4x−3.
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