show that (2x-3) is factor of (2x⁴-x³-3x²-2x+3)
Answers
The division algorithm states that Dividend=Divisor×Quotient+Remainder that is f(x)=g(x)⋅q(x)+r(x)
Here, it is given that the dividend is f(x)=2x
4
−5x
3
+x
2
+3x−2, the divisor is 2x
2
−5x+3 and the remainder is −2x+1, therefore, by applying division algorithm we have:
2x
4
−5x
3
+x
2
+3x−2=(2x
2
−5x+3)g(x)+(−2x+1)
⇒2x
4
−5x
3
+x
2
+3x−2−(−2x+1)=(2x
2
−5x+3)g(x)
⇒2x
4
−5x
3
+x
2
+3x−2+2x−1=(2x
2
−5x+3)g(x)
⇒2x
4
−5x
3
+x
2
+5x−3=(2x
2
−5x+3)g(x)
⇒g(x)=
2x
2
−5x+3
2x
4
−5x
3
+x
2
+5x−3
Let us now divide 2x
4
−5x
3
+x
2
+5x−3 by 2x
2
−5x+3 as shown in the above image:
From the division, we observe that the quotient is x
2
−1 and the remainder is 0.
Hence, the quotient q(x)=x
2
−1.
Answer:
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Step-by-step explanation:
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