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Sum of zeroes =
Product of zeroes =
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Sum of zeroes = −4+(−3)=−71=−(coefficient of x)(coefficient of x2)
Product of zeroes = (−4)(−3)=121=constant termcoefficient of x2
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Page No 48:
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Page No 48:
Answer:
Sum of zeroes = −4+(−3)=−71=−(coefficient of x)(coefficient of x2)
Product of zeroes = (−4)(−3)=121=constant termcoefficient of x2
Page No 48:
Answer:
Page No 48:
Answer:
Sum of zeroes = −4+(−3)=−71=−(coefficient of x)(coefficient of x2)
Product of zeroes = (−4)(−3)=121=constant termcoefficient of x2
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Hence, the relation has been verified.
Page No 49:
Answer:
Sum of zeroes = −4+(−3)=−71=−(coefficient of x)(coefficient of x2)
Product of zeroes = (−4)(−3)=121=constant termcoefficient of x2
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Answer:
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We can find the quadratic equation if we know the sum of the roots and product of the roots by using the formula
x2 − (Sum of the roots)x + Product of roots = 0
Page No 49:
Answer:
Given:
Since, is the root of the above quadratic equation
Hence, It will satisfy the above equation.
Therefore, we will get
Since, is the root of the above quadratic equation
Hence, It will satisfy the above equation.
Therefore, we will get
From (1) and (2), we get
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Answer:
Given: is a factor of
So, we have
Now, It will satisfy the above polynomial.
Therefore, we will get
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Given: is one of the zero of
Now, we have
Now, we divide by to find the quotient
So, the quotient is
Now,
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If the zeroes of the cubic polynomial are a, b and c then the cubic polynomial can be found as
...(1)
Let
Substituting the values in (1), we get
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If the zeroes of the cubic polynomial are a, b and c then the cubic polynomial can be found as
...(1)
Let
Substituting the values in (1), we get
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Answer:
We know the sum, sum of the product of the zeroes taken two at a time and the product of the zeroes of a cubic polynomial then the cubic polynomial can be found as
x3 −(Sum of the zeroes)x2 + (sum of the product of the zeroes taking two at a time)x − Product of zeroes
Therefore, the required polynomial is
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Answer:
Quotient
Remainder
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Quotient
Remainder
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We can write
and
Quotient
Remainder
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Let and
Quotient
Remainder
Since, the remainder is 0.
Hence, is a factor of
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By using division rule, we have
Divided = Quotient × Divisor + Remainder
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We can write and
Quotient =
Remainder =
By using division rule, we have
Divided = Quotient × Divisor + Remainder
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Let the other zeroes of be a.
By using the relationship between the zeroes of the quadratic ploynomial.
We have, Sum of zeroes =
Hence, the other zeroes of is .
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Answer:
By adding and subtracting px, we get
So, the zeros of f(x) are −(p + 1) and p.
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By adding and subtracting mx, we get
So, the zeros of f(x) are −m and m + 3.
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If the zeroes of the quadratic polynomial are α and β then the quadratic polynomial can be found as
x2 − (α + β)x + αβ .....(1)
Substituting the values in (1), we get
x2 − 6x + 4
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Answer:
Given: x = 2 is one zero of the quadratic polynomial kx2 + 3x + k
Therefore, It will satisfy the above polynomial.
Now, we have
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Answer:
Given: x = 3 is one zero of the polynomial 2x2 + x + k
Therefore, It will satisfy the above polynomial.
Now, we have
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Given: x = −4 is one zero of the polynomial x2 − x −(2k + 2)
Therefore, It will satisfy the above polynomial.
Now, we have
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Given: x = 1 is one zero of the polynomial ax2 − 3(a − 1) x − 1
Therefore, It will satisfy the above polynomial.
Now, we have
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Given: x = −2 is one zero of the polynomial 3x2 + 4x + 2k
Therefore, It will satisfy the above polynomial.
Now, we have
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So, the zeros of f(x) are 3 and −2.
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Answer:
By using the relationship between the zeros of the quadratic ploynomial.
We have
Sum of zeroes =
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Answer:
By using the relationship between the zeros of the quadratic ploynomial.
We have
Product of zeroes =
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Given: (x + a) is a factor of 2x2 + 2ax + 5x + 10
We have
Since, (x + a) is a factor of 2x2 + 2ax + 5x + 10
Hence, It will satisfy the above polynomial
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Answer:
By using the relationship between the zeroes of the cubic ploynomial.
We have
Sum of zeroes =
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Answer:
Equating x2 − x to 0 to find the zeros, we will get
Since, x3 + x2 − ax + b is divisible by x2 − x.
Hence, the zeros of x2 − x will satisfy x3 + x2 − ax + b
and
Page No 60:
Answer:
By using the relationship between the zeros of the quadratic ploynomial.
We have,
Sum of zeroes = and Product of zeroes =
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“If f(x) and g(x) are two polynomials such that degree of f(x) is greater than degree of g(x) where g(x) ≠ 0, then there exists unique polynomials q(x) and r(x) such that
f(x) = g(x) × q(x) + r(x),
where
r(
x) = 0 or degree of
r(
x) < degree of
g(
x).
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Answer:
We can find the quadratic polynomial if we know the sum of the roots and product of the roots by using the formula
x2 − (Sum of the zeros)x + Product of zeros
Hence, the required polynomial is .
⇒x2−2√x+13=0⇒3x2−32√x+1=0
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Answer:
To find the zeros of the quadratic polynomial we will equate f(x) to 0
Hence, the zeros of the quadratic polynomial f(x) = 6x2 − 3 are .
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Answer:
To find the zeros of the quadratic polynomial we will equate f(x) to 0
Hence, the zeros of the quadratic polynomial are .
Page No 60:
Answer:
By using the relationship between the zeroes of the quadratic ploynomial.
We have,
Sum of zeroes = and Product of zeroes =
Solving α − β = 1 and α + β = 5, we will get
α = 3 and β = 2
Substituting these values in , we will get
k = 6
Page No 60:
Answer:
By using the relationship between the zeroes of the quadratic ploynomial.
We have,
Sum of zeroes = and Product of zeroes =
Page No 60:
Answer:
By using the relationship between the zeroes of the quadratic ploynomial.
We have,
Sum of zeroes = and Product of zeroes =
Page No 60:
Answer:
By using the relationship between the zeroes of the quadratic ploynomial.
We have,
Sum of zeroes = and Product of zeroes =
Page No 61:
Answer:
By using the relationship between the zeroes of the cubic ploynomial.
We have, Sum of zeroes =
Now, Product of zeros =
Page No 63:
Answer:
(d) is the correct option.
A polynomial in x of degree n is an expression of the form p(x) =ao +a1x+a2x2 +...+an xn, where an 0.
Page No 63:
Answer:
It is because in the second term, the degree of x is −1 and an expression with a negative degree is not a polynomial.
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Multiply by 10, we get
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Page No 69:
Answer:
(b) 3,-1
Here,
Page No 69:
Answer:
(a) −1
Here,
Comparing the given polynomial with , we get:
Page No 69:
Answer:
(c)
Here,
Comparing the given polynomial with , we get:
It is given that are the roots of the polynomial.
Also, =
Page No 69:
Answer:
(c)
Let the zeroes of the polynomial be .
Here, p
Comparing the given polynomial with , we get:
a = 4, b = −8k and c = 9
Now, sum of the roots
Page No 69:
Answer:
Here, p
Page No 69:
Answer:
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Answer:
It is given that the two roots of the polynomial are 2 and −5.
Let
Now, sum of the zeroes, = 2 + (−5) = −3
Product of the zeroes, = 2−5 = −10
∴ Required polynomial =
Page No 69:
Answer:
The given polynomial and its roots are .
Page No 69:
Answer:
Let p
Page No 69:
Answer:
Given:
Sum of the zeroes = −5
Product of the zeroes = 6
∴ Required polynomial =
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Answer:
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Comparing the polynomial with , we get:
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