the polynomial x³+9x²-3x+4 and 9x²-x²+7 when divided by (x-1) leave the reminder R1 and R2 respectively if R2 =2R1 find the value of a
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Step-by-step explanation:
Correct option is
A
2
Let p(x)=x
3
+2x
2
−5ax−7
and q(x)=x
3
+ax
2
−12x+6 be the given polynomials,
Now, R
1
= Remainder when p(x) is divided by x+1.
⇒R
1
=p(−1)
⇒R
1
=(−1)
3
+2(−1)
2
−5a(−1)−7[∵p(x)=x
2
+2x
2
−5ax−7]
⇒R
1
=−1+2+5a−7
⇒R
1
=5a−6
And R
2
= Remainder when q(x) is divided by x-2
⇒R
1
=q(2)
⇒R
2
=(2)
3
+a×2
2
−12×2+6[∵q(x)=x
2
+ax
2
−12x−6]
⇒R
2
=8+4a−24+6
⇒R
2
=4a−10
Substituting the values of R
1
and R
2
in 2R
1
+R
2
=6, we get
⇒2(5a−6)+(4a−10)=6
⇒10a−12+4a−10=6
⇒14a−22=6
⇒14a−28=0
⇒a=2
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