11. Resolve into factors: y4 + y2 + 25
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Answer: hope it helps.
y4+y2+25
y4+y2+25=y4+10y2+25−9y2
y4+y2+25=y4+10y2+25−9y2 =(y2+5)2−9y2
y4+y2+25=y4+10y2+25−9y2 =(y2+5)2−9y2=(y2+5)2−(3y)2
y4+y2+25=y4+10y2+25−9y2 =(y2+5)2−9y2=(y2+5)2−(3y)2=(y2+5+3y)(y2+5−3y)
y4+y2+25=y4+10y2+25−9y2 =(y2+5)2−9y2=(y2+5)2−(3y)2=(y2+5+3y)(y2+5−3y)=(y2+3y+5)(y2−3y+5)
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Given : y⁴ + y² + 25
To Find : Factorize
Solution
y⁴ + y² + 25
Add and subtract 9y²
= y⁴ + y² + 25 + 9y² - 9y²
= y⁴ + 10y² + 5² - (3y)²
= y⁴ + 2.y².5 + 5² - (3y)²
using (a + b)² = a² + 2ab + b²
a = y² b = 5
= (y² + 5)² - (3y)²
using a² - b² = (a + b)(a - b)
a = y² + 5 b = 3y
= (y² + 5 + 3y) (y² + 5 - 3y)
= (y² + 3y + 5)(y² -3y + 5)
y⁴ + y² + 25 = (y² + 3y + 5)(y² -3y + 5)
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