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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.
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The number of zeros in a cubic polynomial is always:
Understanding Zeros of a Cubic Polynomial Mathematical Background A cubic polynomial is of the general form: ax³ + bx² + cx + d, where a ≠ 0 Fundamental Theorem of Algebra - Any polynomial has exactly as many zeros as its degree - These zeros can be real or complex numbers - In a cubic polynomial, tRead more
Understanding Zeros of a Cubic Polynomial
Mathematical Background
A cubic polynomial is of the general form:
ax³ + bx² + cx + d, where a ≠ 0
Fundamental Theorem of Algebra
– Any polynomial has exactly as many zeros as its degree
– These zeros can be real or complex numbers
– In a cubic polynomial, these zeros are referred to as “roots”
Mere Zero Analysis
– A cubic polynomial ALWAYS has 3 zeros
– These zeros can include:
– 3 real zeros in different positions
– 1 real zero and 2 complex conjugate zeros
– 1 repeated real zero that occurs twice
– One real zero that occurs thrice
Mathematical Proof Highlights
– The Fundamental Theorem of Algebra assures 3 zeros
– Complex numbers make all polynomials fully factorable
– Mathematically denoted as:
ax³ + bx² + cx + d = a(x – r₁)(x – r₂)(x – r₃)
Where r₁, r₂, r₃ are the three zeros
Key Insight
The degree of the polynomial always decides the number of zeros,
never mind the actual type of those zeros.
Conclusion:
A cubic polynomial ALWAYS has 3 zeros.
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If one zero of the polynomial x² – 7x + 10 is 5, then the other zero is:
Finding the Second Zero of a Quadratic Polynomial Step 1: Understanding the Given Information - Polynomial: x² – 7x + 10 - One known zero: 5 Step 2: Verification of the Known Zero Let's first verify that 5 is indeed a zero: 5² – 7(5) + 10 = 25 – 35 + 10 = 0 Step 3: Using Vieta's Formulas In a quadraRead more
Finding the Second Zero of a Quadratic Polynomial
Step 1: Understanding the Given Information
– Polynomial: x² – 7x + 10
– One known zero: 5
Step 2: Verification of the Known Zero
Let’s first verify that 5 is indeed a zero:
5² – 7(5) + 10 = 25 – 35 + 10 = 0
Step 3: Using Vieta’s Formulas
In a quadratic polynomial ax² + bx + c, if p and q are zeros:
– Sum of zeros: p + q = -b/a
– Product of zeros: p * q = c/a
For x² – 7x + 10:
– a = 1
– b = -7
– c = 10
Step 4: Finding the Second Zero
We are aware that one zero is 5, therefore let’s use the variable x to represent the second zero.
Sum of zeros formula:
5 + x = 7
x = 7 – 5
x = 2
Verification:
– First zero: 5
– Second zero: 2
– Check sum: 5 + 2 = 7
– Check product: 5 * 2 = 10
Mathematical Insight:
Vieta’s formulas offer a beautiful method of determining polynomial zeros
without resorting to complicated solving methods. They show the profound
connection between a polynomial’s coefficients and its roots.
Conclusion:
The other zero of the polynomial is 2.
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The equation x + 2y = 3 can be written in standard form as:
Step 1: Definition of Standard Form Standard form of a linear equation is represented as: Ax + By = C Where: - A, B, and C are constants - A, B, and C are integers - A and B are not both zero - A ≥ 0 (if A = 0, then B must be positive) Step 2: Given Equation Analysis Original equation: x + 2y = 3 StRead more
Step 1: Definition of Standard Form
Standard form of a linear equation is represented as:
Ax + By = C
Where:
– A, B, and C are constants
– A, B, and C are integers
– A and B are not both zero
– A ≥ 0 (if A = 0, then B must be positive)
Step 2: Given Equation Analysis
Original equation: x + 2y = 3
Step 3: Transformation to Standard Form
– The equation is already very close to standard form
– To make it exactly standard form, we need to move all terms to one side
– Rearrange to: x + 2y – 3 = 0
Verification:
– Coefficients of x: 1
– Coefficients of y: 2
– Constant term: -3
– All parts meet standard form requirements
Step 4: Why Other Options Are Incorrect
– “2y + x = 3” ≠ standard form (terms not on one side)
– “3 – x = 2y” ≠ standard form (rearranged incorrectly)
Conclusion:
The correct standard form is x + 2y – 3 = 0
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If a pair of linear equations is inconsistent, then their graphs will be:
Step 1: Definition of Inconsistent Equations - Inconsistent equations are linear equations that have NO SOLUTION - Graphically, this means the lines representing these equations NEVER intersect Step 2: Graphical Interpretation Inconsistent equations always result in PARALLEL LINES - These lines haveRead more
Step 1: Definition of Inconsistent Equations
– Inconsistent equations are linear equations that have NO SOLUTION
– Graphically, this means the lines representing these equations NEVER intersect
Step 2: Graphical Interpretation
Inconsistent equations always result in PARALLEL LINES
– These lines have the same slope but different y-intercepts
– They run alongside each other, maintaining a constant distance
– No point exists where these lines cross
Step 3: Algebraic Characteristics
Example of Inconsistent Equations:
– Equation ₁: 2x + y = 4
– Equation ₂: 2x + y = 6
Observe:
– Same coefficient for x (2)
– Same coefficient for y (1)
– Different constant terms (4 and 6)
– This guarantees the lines will be parallel
Step 4: Geometric Visualization
– Imagine two identical lines shifted vertically
– They have the same “direction” but never touch
– No matter how far you extend them, they remain parallel
Conclusion:
For a pair of linear equations to be inconsistent, their graphs MUST be PARALLEL LINES.
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