Building Polynomials with Specified Zeros Step 1: Learning Polynomial Building - Provided zeros: -3 and 5 - Simple polynomial form: (x + 3)(x - 5) - Expanding: x² - 2x - 15 Step 2: Freedom Degree Polynomials may be formed by multiplying the simple form by any non-zero constant. Possible Polynomials:Read more
General Form:
For any non-zero constant k:
k(x² – 2x – 15)
Mathematical Insight:
– The k is the parameter for infinite scaling of the polynomial
– Each scale produces a different polynomial with the same zeros
– These are actually the same polynomial to a constant factor
Step 3: Counting Polynomials
– We can make polynomials with these zeros by multiplying the fundamental form
by any real number
– This produces an infinite number of polynomials
Conclusion
There are MORE THAN 3 polynomials with zeros -3 and 5.
Building a Quadratic Polynomial with Given Zeros Step 1: Defining Zeros - Given zeros: 4 and -3 - The polynomial shall thus be in the form: (x - 4)(x + 3) Step 2: Algebraic Expansion (x - 4)(x + 3) = x² + 3x - 4x - 12 = x² - x - 12 Step 3: Checking Zero Properties Let's test if the zeros hold: - WheRead more
Building a Quadratic Polynomial with Given Zeros
Step 1: Defining Zeros
– Given zeros: 4 and -3
– The polynomial shall thus be in the form: (x – 4)(x + 3)
Step 3: Checking Zero Properties
Let’s test if the zeros hold:
– When x = 4:
4² – 4 – 12 = 16 – 4 – 12 = 0 OK
– When x = -3:
(-3)² – (-3) – 12 = 9 + 3 – 12 = 0 OK
Step 4: Coefficient Analysis
– Coefficient of x²: 1
– Coefficient of x: -1
– Constant term: -12
Mathematical Insights:
– The zeros of a quadratic define its form
– The general form illustrates how the zeros are connected to the coefficients of the polynomial
– Vieta’s formulas support the connection between coefficients and zeros
Conclusion:
The quadratic polynomial whose zeros are 4 and -3 is x² – x – 12.
Solving for the Unknown Zero of a Quadratic Polynomial Step 1: Given Information - Polynomial: 2x² + 7x + 3 - One known zero: -3/2 Step 2: Vieta's Formulas for Quadratic Polynomials Let the two zeros be p and q: - p + q = -b/a - p * q = c/a Where in 2x² + 7x + 3: - a = 2 - b = 7 - c = 3 Step 3: UsinRead more
Solving for the Unknown Zero of a Quadratic Polynomial
Step 1: Given Information
– Polynomial: 2x² + 7x + 3
– One known zero: -3/2
Step 2: Vieta’s Formulas for Quadratic Polynomials
Let the two zeros be p and q:
– p + q = -b/a
– p * q = c/a
Where in 2x² + 7x + 3:
– a = 2
– b = 7
– c = 3
Step 3: Using the Known Zero
If p = -3/2, then we need to find q
Sum of zeros formula:
p + q = -b/a
-3/2 + q = -7/2
Step 4: Solving for the Unknown Zero
q = -7/2 – (-3/2)
= -7/2 + 3/2
= -4/2
= -2
Verification:
– First zero: p = -3/2
– Second zero: q = -2
– Check: (-3/2) + (-2) = -7/2 ✓
– Check: (-3/2) * (-2) = 3/a ✓
Mathematical Insight:
Vieta’s formulas provide a powerful method to find
polynomial zeros without fully solving the equation.
Conclusion:
The other zero of the polynomial is -2 (or 1 in the given options).
Determining the Remainder of Polynomial Division Step 1: Remainder Theorem Basics - The remainder theorem says that when a polynomial f(x) is divided by (x - a), the remainder is f(a) - This implies we can determine the remainder by evaluating the polynomial at x = 2 Step 2: Evaluating a PolynomialRead more
Determining the Remainder of Polynomial Division
Step 1: Remainder Theorem Basics
– The remainder theorem says that when a polynomial f(x) is divided by (x – a),
the remainder is f(a)
– This implies we can determine the remainder by evaluating the polynomial at x = 2
Step 2: Evaluating a Polynomial
Polynomial: f(x) = x³ – 6x² + 11x – 6
Substituting x = 2:
Mathematical Insight:
– By simply putting the root of the divisor (2) in the polynomial
– We can easily find the remainder without long division
– It is a very strong method that makes polynomial remainder computation easier
Step 3: Confirmation
– The remainder is 0
– It indicates (x – 2) evenly divides the polynomial
– There is no remaind value when divided
Conclusion:
The remainder upon division of x³ – 6x² + 11x – 6 by (x – 2) is zero.
Constructing a Quadratic Polynomial with Specific Zero Properties Step 1: Understanding Vieta's Formulas For a quadratic polynomial ax² + bx + c with zeros p and q: - Sum of zeros: p + q = -b/a - Product of zeros: p * q = c/a Given Conditions: - Sum of zeros = -4 - Product of zeros = 3 Step 2: AnalyRead more
Constructing a Quadratic Polynomial with Specific Zero Properties
Step 1: Understanding Vieta’s Formulas
For a quadratic polynomial ax² + bx + c with zeros p and q:
– Sum of zeros: p + q = -b/a
– Product of zeros: p * q = c/a
Given Conditions:
– Sum of zeros = -4
– Product of zeros = 3
Step 2: Analyzing the Coefficients
Let’s consider a standard quadratic form: x² + 4x + c
Checking Sum of Zeros:
– p + q = -4
– This means the coefficient of x must be -4
Checking Product of Zeros:
– p * q = 3
– This means the constant term must be 3
Checking Conditions:
– Sum of zeros: 3 + 1 = 4 ✓
– Product of zeros: 3 * 1 = 3 ✓
Key Insights:
– Vieta’s formulas provide a powerful way to relate
zeros to polynomial coefficients
– We can construct polynomials by understanding
the relationships between zeros and coefficients
Conclusion:
The polynomial that satisfies the given conditions is x² – 4x + 3.
The number of polynomials having zeros -3 and 5 is
Building Polynomials with Specified Zeros Step 1: Learning Polynomial Building - Provided zeros: -3 and 5 - Simple polynomial form: (x + 3)(x - 5) - Expanding: x² - 2x - 15 Step 2: Freedom Degree Polynomials may be formed by multiplying the simple form by any non-zero constant. Possible Polynomials:Read more
Building Polynomials with Specified Zeros
Step 1: Learning Polynomial Building
– Provided zeros: -3 and 5
– Simple polynomial form: (x + 3)(x – 5)
– Expanding: x² – 2x – 15
Step 2: Freedom Degree
Polynomials may be formed by multiplying the simple form by any non-zero constant.
Possible Polynomials:
1. x² – 2x – 15
2. 2x² – 4x – 30
3. 3x² – 6x – 45
General Form:
For any non-zero constant k:
k(x² – 2x – 15)
Mathematical Insight:
– The k is the parameter for infinite scaling of the polynomial
– Each scale produces a different polynomial with the same zeros
– These are actually the same polynomial to a constant factor
Step 3: Counting Polynomials
– We can make polynomials with these zeros by multiplying the fundamental form
by any real number
– This produces an infinite number of polynomials
Conclusion
There are MORE THAN 3 polynomials with zeros -3 and 5.
Click here for more:
See lesshttps://www.tiwariacademy.in/ncert-solutions/class-10/maths/
The quadratic polynomial whose zeros are 4 and -3 is:
Building a Quadratic Polynomial with Given Zeros Step 1: Defining Zeros - Given zeros: 4 and -3 - The polynomial shall thus be in the form: (x - 4)(x + 3) Step 2: Algebraic Expansion (x - 4)(x + 3) = x² + 3x - 4x - 12 = x² - x - 12 Step 3: Checking Zero Properties Let's test if the zeros hold: - WheRead more
Building a Quadratic Polynomial with Given Zeros
Step 1: Defining Zeros
– Given zeros: 4 and -3
– The polynomial shall thus be in the form: (x – 4)(x + 3)
Step 2: Algebraic Expansion
(x – 4)(x + 3) = x² + 3x – 4x – 12
= x² – x – 12
Step 3: Checking Zero Properties
Let’s test if the zeros hold:
– When x = 4:
4² – 4 – 12 = 16 – 4 – 12 = 0 OK
– When x = -3:
(-3)² – (-3) – 12 = 9 + 3 – 12 = 0 OK
Step 4: Coefficient Analysis
– Coefficient of x²: 1
– Coefficient of x: -1
– Constant term: -12
Mathematical Insights:
– The zeros of a quadratic define its form
– The general form illustrates how the zeros are connected to the coefficients of the polynomial
– Vieta’s formulas support the connection between coefficients and zeros
Conclusion:
The quadratic polynomial whose zeros are 4 and -3 is x² – x – 12.
Click here for more:
See lesshttps://www.tiwariacademy.in/ncert-solutions/class-10/maths/
If one of the zeros of the polynomial 2x² + 7x + 3 is -3/2, then the other zero is:
Solving for the Unknown Zero of a Quadratic Polynomial Step 1: Given Information - Polynomial: 2x² + 7x + 3 - One known zero: -3/2 Step 2: Vieta's Formulas for Quadratic Polynomials Let the two zeros be p and q: - p + q = -b/a - p * q = c/a Where in 2x² + 7x + 3: - a = 2 - b = 7 - c = 3 Step 3: UsinRead more
Solving for the Unknown Zero of a Quadratic Polynomial
Step 1: Given Information
– Polynomial: 2x² + 7x + 3
– One known zero: -3/2
Step 2: Vieta’s Formulas for Quadratic Polynomials
Let the two zeros be p and q:
– p + q = -b/a
– p * q = c/a
Where in 2x² + 7x + 3:
– a = 2
– b = 7
– c = 3
Step 3: Using the Known Zero
If p = -3/2, then we need to find q
Sum of zeros formula:
p + q = -b/a
-3/2 + q = -7/2
Step 4: Solving for the Unknown Zero
q = -7/2 – (-3/2)
= -7/2 + 3/2
= -4/2
= -2
Verification:
– First zero: p = -3/2
– Second zero: q = -2
– Check: (-3/2) + (-2) = -7/2 ✓
– Check: (-3/2) * (-2) = 3/a ✓
Mathematical Insight:
Vieta’s formulas provide a powerful method to find
polynomial zeros without fully solving the equation.
Conclusion:
The other zero of the polynomial is -2 (or 1 in the given options).
Click here for more:
See lesshttps://www.tiwariacademy.in/ncert-solutions/class-10/maths/
If the polynomial x³ – 6x² + 11x – 6 is divided by (x – 2), then the remainder is:
Determining the Remainder of Polynomial Division Step 1: Remainder Theorem Basics - The remainder theorem says that when a polynomial f(x) is divided by (x - a), the remainder is f(a) - This implies we can determine the remainder by evaluating the polynomial at x = 2 Step 2: Evaluating a PolynomialRead more
Determining the Remainder of Polynomial Division
Step 1: Remainder Theorem Basics
– The remainder theorem says that when a polynomial f(x) is divided by (x – a),
the remainder is f(a)
– This implies we can determine the remainder by evaluating the polynomial at x = 2
Step 2: Evaluating a Polynomial
Polynomial: f(x) = x³ – 6x² + 11x – 6
Substituting x = 2:
f(2) = 2³ – 6(2)² + 11(2) – 6
= 8 – 6(4) + 22 – 6
= 8 – 24 + 22 – 6
= 0
Mathematical Insight:
– By simply putting the root of the divisor (2) in the polynomial
– We can easily find the remainder without long division
– It is a very strong method that makes polynomial remainder computation easier
Step 3: Confirmation
– The remainder is 0
– It indicates (x – 2) evenly divides the polynomial
– There is no remaind value when divided
Conclusion:
The remainder upon division of x³ – 6x² + 11x – 6 by (x – 2) is zero.
Click here for more:
See lesshttps://www.tiwariacademy.in/ncert-solutions/class-10/maths/
If the sum of the zeros of the quadratic polynomial ax² + bx + c is -4 and the product of the zeros is 3, then the polynomial is:
Constructing a Quadratic Polynomial with Specific Zero Properties Step 1: Understanding Vieta's Formulas For a quadratic polynomial ax² + bx + c with zeros p and q: - Sum of zeros: p + q = -b/a - Product of zeros: p * q = c/a Given Conditions: - Sum of zeros = -4 - Product of zeros = 3 Step 2: AnalyRead more
Constructing a Quadratic Polynomial with Specific Zero Properties
Step 1: Understanding Vieta’s Formulas
For a quadratic polynomial ax² + bx + c with zeros p and q:
– Sum of zeros: p + q = -b/a
– Product of zeros: p * q = c/a
Given Conditions:
– Sum of zeros = -4
– Product of zeros = 3
Step 2: Analyzing the Coefficients
Let’s consider a standard quadratic form: x² + 4x + c
Checking Sum of Zeros:
– p + q = -4
– This means the coefficient of x must be -4
Checking Product of Zeros:
– p * q = 3
– This means the constant term must be 3
Step 3: Verification
The polynomial becomes: x² – 4x + 3
Mathematical Verification:
Let’s find the zeros using the quadratic formula:
x = [4 ± √(16 – 4(1)(3))] / 2(1)
= [4 ± √(16 – 12)] / 2
= [4 ± √4] / 2
= [4 ± 2] / 2
Zeros are:
– p = (4 + 2)/2 = 3
– q = (4 – 2)/2 = 1
Checking Conditions:
– Sum of zeros: 3 + 1 = 4 ✓
– Product of zeros: 3 * 1 = 3 ✓
Key Insights:
– Vieta’s formulas provide a powerful way to relate
zeros to polynomial coefficients
– We can construct polynomials by understanding
the relationships between zeros and coefficients
Conclusion:
The polynomial that satisfies the given conditions is x² – 4x + 3.
Click here for more:
See lesshttps://www.tiwariacademy.in/ncert-solutions/class-10/maths/