If the polynomials 2x + ar? + 3x – 5 and x + x2 - 4x + a leave the same rest
when divided by X-2. find the value of a
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Answer:
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Let the given polynomials be f(x) and g(x).
ATQ,
When f(x) and g(x) are divided by (x-2) they leave the same remainder.
I.e (x-2) is a factor of f(x) and g(x). It means 2 is the zero of f(x) and g(x)
So that,
f(2) = g(2)
2x³+ax²+3x-5 = x³+x²-4x+a
2(2)³+a(2)²+3(2)-5 = 2³+2²-4(2)+a
2(8)+a(4)+6-5 = 8+4-8+a
16+4a+1 = 4+a
17+4a = 4+a
4a-a = 4-17
3a = -13
a = -13/3.
Hope helped!
:)
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Let p(x) = 2x3 + ax2 + 3x – 5 and q(x) = x3 + x2 – 4x + a
As we know by Remainder Theorem,
If a polynomial p(x) is divided by a linear polynomial (x – a) then, the remainder is p(a)
⇒ Remainder of p(x) when divided by x – 2 is p(2). Similarly, Remainder of q(x) when divided by x – 2 is q(2)
⇒ p(2) = 2(2)3 +a(2)2 + 3(2) – 5
⇒p(2) = 16 + 4a +6 – 5
⇒p(2) = 17 + 4a
Similarly, q(2) = (2)3 + (2)2 + –4(2) + a
⇒ q(2) = 8 + 4 –8 + a
⇒ q(2) = 4 + a
Since they both leave the same remainder, so p(2) = q(2)
⇒ 17 + 4a = 4 + a
⇒ 13 = -3a
∴ The value of a is –13/3
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Let the given polynomials be f(x) and g(x).
ATQ,
When f(x) and g(x) are divided by (x-2) they leave the same remainder.
I.e (x-2) is a factor of f(x) and g(x). It means 2 is the zero of f(x) and g(x)
So that,
f(2) = g(2)
2x³+ax²+3x-5 = x³+x²-4x+a
2(2)³+a(2)²+3(2)-5 = 2³+2²-4(2)+a
2(8)+a(4)+6-5 = 8+4-8+a
16+4a+1 = 4+a
17+4a = 4+a
4a-a = 4-17
3a = -13
a = -13/3.
Hope helped!
:)
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