H Algebra 2/Trig · Unit 0
Let u = x². An x⁴ trinomial becomes a quadratic in u.
U-Substitution
Use u-substitution when the polynomial is ax⁴ + bx² + c. The powers are x⁴, x², and a constant, so the trinomial is quadratic in x².
ax⁴ + bx² + c
u = x²
a(x²)² + b(x²) + c
au² + bu + c
Factor the quadratic in u. Then replace u with x².
When the number in front of u² is 1, find two numbers that multiply to the constant and add to the middle coefficient.
When that number is greater than 1, factor the u-quadratic with Slide and Divide.
Factor. x⁴ + 5x² + 6
u = x²
(x²)² + 5(x²) + 6
u² + 5u + 6
Find two numbers that multiply to 6 and add to 5.
2 and 3:
2*3 = 6 and 2+3 = 5.
(u+2)(u+3)
Replace u with x².
(x²+2)(x²+3)
The number in front of x⁴ is 1.
Factor. x⁴ + 5x² + 6
u = x²
(x²)² + 5(x²) + 6
u² + 5u + 6
Find two numbers that multiply to 6 and add to 5.
2 and 3:
2*3 = 6 and 2+3 = 5.
(u+2)(u+3)
Replace u with x².
(x²+2)(x²+3)
Answer: (x²+2)(x²+3)
Factor. x⁴ + 9x² + 14
u = x²
(x²)² + 9(x²) + 14
u² + 9u + 14
Find two numbers that multiply to 14 and add to 9.
2 and 7:
2*7 = 14 and 2+7 = 9.
(u+2)(u+7)
Replace u with x².
(x²+2)(x²+7)
Answer: (x²+2)(x²+7)
Factor. x⁴ + 10x² + 21
u = x²
(x²)² + 10(x²) + 21
u² + 10u + 21
Find two numbers that multiply to 21 and add to 10.
3 and 7:
3*7 = 21 and 3+7 = 10.
(u+3)(u+7)
Replace u with x².
(x²+3)(x²+7)
Answer: (x²+3)(x²+7)
Factor. x⁴ − 8x² + 15
u = x²
(x²)² − 8(x²) + 15
u² − 8u + 15
Find two numbers that multiply to 15 and add to −8.
−3 and −5:
−3*−5 = 15 and −3+(−5) = −8.
(u−3)(u−5)
Replace u with x².
(x²−3)(x²−5)
Answer: (x²−3)(x²−5)
Factor. x⁴ + 2x² − 15
u = x²
(x²)² + 2(x²) − 15
u² + 2u − 15
Find two numbers that multiply to −15 and add to 2.
−3 and 5:
−3*5 = −15 and −3+5 = 2.
(u−3)(u+5)
Replace u with x².
(x²−3)(x²+5)
Answer: (x²−3)(x²+5)
Factor. x⁴ − 4x² − 12
u = x²
(x²)² − 4(x²) − 12
u² − 4u − 12
Find two numbers that multiply to −12 and add to −4.
2 and −6:
2*−6 = −12 and 2+(−6) = −4.
(u+2)(u−6)
Replace u with x².
(x²+2)(x²−6)
Answer: (x²+2)(x²−6)
The number in front of x⁴ is greater than 1. Factor the u-quadratic with Slide and Divide.
Factor. 6x⁴ + 11x² + 3
u = x²
6(x²)² + 11(x²) + 3
6u² + 11u + 3
Factor the u-quadratic with Slide and Divide.
SLIDE, multiply 6*3
6u² + 11u + 3
Solve the Puzzle. What two numbers multiply to 18 and add to 11.
1*18
2*9
3*6
6u²
+ 3
Now we divided 11u into two pieces 2u and 9u.
6u² + 2u + 9u + 3
Factor by grouping.
6u² + 2u + 9u + 3
Factor the GCF from each part.
2u(3u+1) + 3(3u+1)
(3u+1)(2u+3)
Replace u with x².
(3x²+1)(2x²+3)
Answer: (3x²+1)(2x²+3)
Factor. 9x⁴ − 9x² + 2
u = x²
9(x²)² − 9(x²) + 2
9u² − 9u + 2
Factor the u-quadratic with Slide and Divide.
SLIDE, multiply 9*2
9u² − 9u + 2
Solve the Puzzle. What two numbers multiply to 18 and add to −9.
(−1)*(−18)
(−2)*(−9)
(−3)*(−6)
9u²
+ 2
Now we divided −9u into two pieces −3u and −6u.
9u² − 3u − 6u + 2
Factor by grouping.
9u² − 3u − 6u + 2
Factor the GCF from each part.
3u(3u−1) − 2(3u−1)
(3u−1)(3u−2)
Replace u with x².
(3x²−1)(3x²−2)
Answer: (3x²−1)(3x²−2)
Factor. 8x⁴ − 14x² + 5
u = x²
8(x²)² − 14(x²) + 5
8u² − 14u + 5
Factor the u-quadratic with Slide and Divide.
SLIDE, multiply 8*5
8u² − 14u + 5
Solve the Puzzle. What two numbers multiply to 40 and add to −14.
(−1)*(−40)
(−2)*(−20)
(−4)*(−10)
(−5)*(−8)
8u²
+ 5
Now we divided −14u into two pieces −4u and −10u.
8u² − 4u − 10u + 5
Factor by grouping.
8u² − 4u − 10u + 5
Factor the GCF from each part.
4u(2u−1) − 5(2u−1)
(2u−1)(4u−5)
Replace u with x².
(2x²−1)(4x²−5)
Answer: (2x²−1)(4x²−5)
Factor. 2x⁴ − 7x² − 15
u = x²
2(x²)² − 7(x²) − 15
2u² − 7u − 15
Factor the u-quadratic with Slide and Divide.
SLIDE, multiply 2*(−15)
2u² − 7u − 15
Solve the Puzzle. What two numbers multiply to −30 and add to −7.
(−1)*30
1*(−30)
(−2)*15
2*(−15)
(−3)*10
3*(−10)
(−5)*6
5*(−6)
2u²
− 15
Now we divided −7u into two pieces 3u and −10u.
2u² + 3u − 10u − 15
Factor by grouping.
2u² + 3u − 10u − 15
Factor the GCF from each part.
u(2u+3) − 5(2u+3)
(2u+3)(u−5)
Replace u with x².
(2x²+3)(x²−5)
Answer: (2x²+3)(x²−5)
Factor. 4x⁴ + 17x² − 15
u = x²
4(x²)² + 17(x²) − 15
4u² + 17u − 15
Factor the u-quadratic with Slide and Divide.
SLIDE, multiply 4*(−15)
4u² + 17u − 15
Solve the Puzzle. What two numbers multiply to −60 and add to 17.
(−1)*60
1*(−60)
(−2)*30
2*(−30)
(−3)*20
3*(−20)
(−4)*15
4*(−15)
(−5)*12
5*(−12)
(−6)*10
6*(−10)
4u²
− 15
Now we divided 17u into two pieces −3u and 20u.
4u² − 3u + 20u − 15
Factor by grouping.
4u² − 3u + 20u − 15
Factor the GCF from each part.
u(4u−3) + 5(4u−3)
(4u−3)(u+5)
Replace u with x².
(4x²−3)(x²+5)
Answer: (4x²−3)(x²+5)
Factor. 10x⁴ + 13x² − 3
u = x²
10(x²)² + 13(x²) − 3
10u² + 13u − 3
Factor the u-quadratic with Slide and Divide.
SLIDE, multiply 10*(−3)
10u² + 13u − 3
Solve the Puzzle. What two numbers multiply to −30 and add to 13.
(−1)*30
1*(−30)
(−2)*15
2*(−15)
(−3)*10
3*(−10)
(−5)*6
5*(−6)
10u²
− 3
Now we divided 13u into two pieces −2u and 15u.
10u² − 2u + 15u − 3
Factor by grouping.
10u² − 2u + 15u − 3
Factor the GCF from each part.
2u(5u−1) + 3(5u−1)
(5u−1)(2u+3)
Replace u with x².
(5x²−1)(2x²+3)
Answer: (5x²−1)(2x²+3)
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