Without actual division, prove that 2x^4 – 5x^3 + 2x^2 – x + 2 is divisible by x^2 – 3x + 2.
We use the Factor Theorem. Factor x^2 – 3x + 2 first:
x^2 – 3x + 2 = (x – 1)(x – 2)
If the given polynomial P(x) is divisible by x^2 – 3x + 2, then both x = 1 and x = 2 should satisfy P(x) = 0.
Let P(x) = 2x^4 – 5x^3 + 2x^2 – x + 2
1. Check x = 1:
P(1) = 2(1)^4 – 5(1)^3 + 2(1)^2 – 1 + 2
= 2 – 5 + 2 – 1 + 2
= 0
2. Check x = 2:
P(2) = 2(16) – 5(8) + 2(4) – 2 + 2
= 32 – 40 + 8 – 2 + 2
= 0
Since P(1) = 0 and P(2) = 0, the polynomial is divisible by (x – 1)(x – 2) = x^2 – 3x + 2.PolynomialsMore Questions on Polynomials
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