If x+5 is a factor of 2x^3-4x^2+ax+5, then value of a is
the remainder when f(x) is divided by (x+5) is f(-5) = -5a-345
so, since x+5 is a factor, that means that
-5a-345=0
a = -69
check:
2x^3 - 4x^2 - 69x + 5 = (x+5)(2x^2 - 14x + 1)
Tatakae
To find the value of 'a', we can use the factor theorem. According to the factor theorem, if (x + 5) is a factor of the given polynomial, then substituting -5 for 'x' in the polynomial should result in zero.
Let's substitute -5 for 'x' in the polynomial 2x^3 - 4x^2 + ax + 5:
2(-5)^3 - 4(-5)^2 + a(-5) + 5
= 2(-125) - 4(25) - 5a + 5
= -250 - 100 - 5a + 5
= -350 - 5a + 5
Simplifying further:
= -345 - 5a
Since the polynomial should equal zero when (x + 5) is a factor, we set the equation equal to zero:
-345 - 5a = 0
To solve for 'a', we isolate it by bringing -345 to the other side:
-5a = 345
Dividing both sides by -5, we get:
a = -345 / -5
a = 69
So, the value of 'a' is 69.
To find the value of 'a' when x+5 is a factor of 2x^3-4x^2+ax+5, we can use the remainder theorem. The remainder theorem states that if 'a' is a factor of a polynomial, then when dividing the polynomial by 'a', the remainder will be zero.
In this case, since x+5 is a factor, we can say that when dividing 2x^3-4x^2+ax+5 by x+5, the remainder will be zero.
To find the remainder, we can use polynomial long division. Let's divide 2x^3-4x^2+ax+5 by x+5:
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x+5 | 2x^3 - 4x^2 + ax + 5
To start the division, we divide the first term, 2x^3, by x, which gives us 2x^2. We write it above the line.
2x^2
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x+5 | 2x^3 - 4x^2 + ax + 5
Next, we multiply this result (2x^2) by (x+5) and subtract it from the original polynomial:
2x^2
-2x^2 - 10x
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x+5 | 2x^3 - 4x^2 + ax + 5
- (2x^3 + 10x^2)
The terms (2x^3 - 2x^3) cancel out, leaving only (-4x^2 - 10x) in the second line.
Now, we bring down the next term, which is 'ax':
2x^2 - 10x
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x+5 | 2x^3 - 4x^2 + ax + 5
- (2x^3 + 10x^2)
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-14x^2 + ax
Next, we divide (-14x^2 + ax) by (x+5).
Dividing the leading term, -14x^2, by x gives us -14x. We write it above the line.
2x^2 - 10x - 14x
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x+5 | 2x^3 - 4x^2 + ax + 5
- (2x^3 + 10x^2)
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-14x^2 + ax
- (-14x^2 - 70x)
Again, the terms (-14x^2 - (-14x^2)) cancel out, leaving only (ax - 70x) in the second line.
Now, we bring down the last term, which is '5':
2x^2 - 10x - 14x
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x+5 | 2x^3 - 4x^2 + ax + 5
- (2x^3 + 10x^2)
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-14x^2 + ax
- (-14x^2 - 70x)
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ax - 70x + 5
Now, since we have reached the end of the polynomial, we can see that the remainder is given by (ax - 70x + 5).
For this remainder to be zero, the coefficient of 'x' should be zero.
Therefore, we can set ax - 70x + 5 = 0 and solve for 'a':
ax - 70x + 5 = 0
Simplifying the equation:
(a - 70)x + 5 = 0
Since the expression is equal to zero, the coefficient of 'x' should be zero:
a - 70 = 0
Solving for 'a':
a = 70
So, the value of 'a' is 70.