Polynomials

Let the given polynomial be .
We are given that and are two zeros of the polynomial .

By the definition of a zero of a polynomial, if is a zero of , then .
Since is a zero, we must have .
Substituting into the polynomial expression:




--- (Equation 1)

Since is also a zero, we must have .
Substituting into the polynomial expression:




Divide the entire equation by 2 to simplify:

--- (Equation 2)

Now we need to solve the system of two linear equations with two variables, and :
1)
2)

To eliminate , we can subtract Equation 1 from Equation 2:

Substitute the value back into Equation 1:

Therefore, the values of and are and .
The required pair is .

The correct option is (B):

Step 1: Recall the relationship between degrees during polynomial division.

When dividing two polynomials, the degree of the quotient is given by:

Step 2: Apply the formula.

The degree of the dividend is 7, and the degree of the divisor is 4. Thus:

Final Answer: The degree of the quotient is , which corresponds to option .

We are tasked with finding the value of , where and are the zeros of the quadratic polynomial .

Step 1: Recall the relationships between the roots and coefficients of a quadratic polynomial.

For a quadratic polynomial , the sum and product of the roots are given by:

Here, the polynomial is , so , , and . Thus:

Step 2: Simplify .

Using the formula for the sum of reciprocals:

Substitute the values of and :

Final Answer: The value of is , which corresponds to option .

Step 1: Recall the Factor Theorem.

The Factor Theorem states that if is a factor of a polynomial , then . To check if any of the given options is a factor, we substitute the corresponding value of into and verify if .

Step 2: Test each option.

(1) :

If is a factor, then . Substitute into :

Since , is not a factor.

(2) :

If is a factor, then . Substitute into :

Since , is a factor.

(3) :

If is a factor, then . Substitute into :

Simplify step-by-step:

Convert all terms to have a denominator of 8:

Since , is not a factor.

(4) :

If is a factor, then . Substitute into :

Simplify step-by-step:

Convert all terms to have a denominator of 8:

Since , is not a factor.

Final Answer: The correct factor is , which corresponds to option .

Correct answer: quadratic equation

Explanation:
An equation of the form , where is a polynomial of degree 2, is called a quadratic equation.
The general form of a quadratic equation is:

Linear equations are of degree 1, not 2. So, the other options are incorrect.

Correct answer:

Explanation:
The general form of a quadratic polynomial whose zeroes are α and β is:



Here,
- is the sum of the zeroes
- is the product of the zeroes

Hence, the correct form is:

Correct answer: 0, 0, 1

Explanation:
Given polynomial: x³ – x²

Factor the expression:
x²(x – 1)

Now, set the polynomial equal to 0:
x²(x – 1) = 0

So the zeroes are:
x = 0 (with multiplicity 2), and x = 1

Hence, the zeroes are: 0, 0, 1

Correct answer: 4

Explanation:
A biquadratic polynomial is a polynomial of degree 4.

According to the Fundamental Theorem of Algebra, a polynomial of degree n can have at most n real or complex zeroes.

Therefore, a biquadratic polynomial can have at most 4 zeroes.

The number of zeroes of a polynomial is determined by the number of times the graph intersects the x-axis. In the given graph, the polynomial intersects the x-axis at three points. Therefore, the polynomial has 3 zeroes.

The correct option is (D): 3

We are given the polynomial: To find the degree of the polynomial, we will first expand it. The degree of the polynomial is determined by the highest power of after expansion. Let’s expand the expression: When we multiply the terms, we get: Now, combine like terms: The highest power of is , so the degree of the polynomial is 5.

The correct option is (D):

We are asked to identify which of the following is not a polynomial.
is a polynomial because all the exponents of are non-negative integers.
is a polynomial because all the exponents of are non-negative integers.
is a polynomial because all the exponents of are non-negative integers.
is not a polynomial because it contains a negative exponent of ( ).
Thus, the expression is not a polynomial.

The correct option is (D):

The number of zeroes of a polynomial is determined by the number of points where its graph intersects the x-axis.
Step 1: Identify the zeroes by observing the points where the given graph touches or crosses the x-axis.
Step 2: Count the number of intersections.
Conclusion: If the graph does not intersect the x-axis, the polynomial has 0 zeroes.
If it intersects once, it has 1 zero. If it intersects twice, it has 2 zeroes.

Given the quadratic polynomial f(x) = 6x² + x - 2 with zeroes α and β.

For any quadratic polynomial of the form f(x) = ax² + bx + c:

• Sum of zeroes (α + β) = -b/a
• Product of zeroes (αβ) = c/a

Applying this to our polynomial:

• a = 6 (coefficient of x²)
• b = 1 (coefficient of x)
• c = -2 (constant term)

Calculating the sum of zeroes:

α + β = -b/a = -1/6

Therefore, the sum of the zeroes is -1/6.

The correct answer is: (2) -1/6

Given the cubic polynomial 4x³ - 6x² + 7x + 3 = 0 with roots α, β, γ.

For any cubic polynomial of the form ax³ + bx² + cx + d = 0, the following relationships hold for its roots:

• Sum of roots (α + β + γ) = -b/a
• Sum of product of roots taken two at a time (αβ + βγ + γα) = c/a
• Product of roots (αβγ) = -d/a

For the given polynomial 4x³ - 6x² + 7x + 3 = 0:

• a = 4 (coefficient of x³)
• b = -6 (coefficient of x²)
• c = 7 (coefficient of x)
• d = 3 (constant term)

We need to find the value of αβ + βγ + γα, which is given by c/a.

Calculating this:

αβ + βγ + γα = c/a = 7/4

Therefore, the value of αβ + βγ + γα is 7/4.

The correct answer is: (2) 7/4

Given the quadratic polynomial ax² + bx + c (c ≠ 0) with equal zeroes.

For a quadratic polynomial to have equal zeroes, its discriminant must be zero:

Discriminant (D) = b² - 4ac = 0

This implies: b² = 4ac

Now considering the product of roots (αα = α²):

Product of roots = c/a = α²

Since c ≠ 0 and α² > 0 (as squares are always positive for real numbers), we conclude:

• c/a must be positive
• This means c and a must have the same sign

Analyzing the options:

1. c and a have opposite signs → False
2. c and a have the same signs → True
3. b² ≠ 4ac → False (as b² = 4ac for equal roots)
4. None of these → False

Therefore, the correct answer is: (2) c and a have the same signs

A polynomial must satisfy three conditions:

1. Variables must have non-negative integer exponents
2. No variables in denominators
3. No variables under roots

Option (1): x² - 6√x + 2
This fails condition 3 because √x = x^1/2 has a fractional exponent.

Option (2): 5/(x⁴ - 3x + 1)
This fails condition 2 because variables appear in the denominator.

Option (3): 5x² - 3x + √2
This satisfies all conditions:

• Exponents are integers (2, 1, 0)
• √2 is a constant coefficient
• No variables in denominators or roots

Option (4): 2x² - 5/x + 3
This fails condition 2 because 5/x = 5x^-1 has a negative exponent.

Therefore, the correct answer is: (3) 5x² - 3x + √2