All questions of Polynomials for ACT Exam

To determine the value of k in the polynomial f(x) = x^2 + kx - 15, we need to check if the polynomial is exactly divisible by x - 5.

To check if the polynomial is divisible by x - 5, we can use the remainder theorem. According to the remainder theorem, if a polynomial f(x) is divided by x - a and the remainder is zero, then f(a) = 0.

So, let's substitute x = 5 into the polynomial and check if the result is zero.

Substituting x = 5 into f(x), we get:
f(5) = (5)^2 + k(5) - 15
= 25 + 5k - 15
= 5k + 10

If the polynomial is exactly divisible by x - 5, then f(5) should be equal to zero.

Setting f(5) = 0, we have:
5k + 10 = 0

Solving for k, we get:
5k = -10
k = -10/5
k = -2

Therefore, the value of k is -2.

Hence, the correct answer is option 'D' (-2).

Explanation:
There are 3 terms in the polynomial 5x^2 + 2x - 2.

Term 1: 5x^2
- This is a term because it consists of a coefficient (5) multiplied by a variable raised to a power (x^2).

Term 2: 2x
- This is a term because it consists of a coefficient (2) multiplied by a variable (x).

Term 3: -2
- This is a term because it is a constant term, which means it is not multiplied by any variable.
Therefore, the polynomial has a total of 3 terms. Each term is separated by a plus or minus sign.

The degree of a polynomial is the highest power of the variable in the polynomial. In this case, the polynomial is 4x^7 + 9x^5 + 5x^2 + 11.

To determine the degree of this polynomial, we need to look for the term with the highest power of x.

- Identify the terms and their degrees
The polynomial consists of four terms: 4x^7, 9x^5, 5x^2, and 11. The degrees of these terms are as follows:
- 4x^7 has a degree of 7
- 9x^5 has a degree of 5
- 5x^2 has a degree of 2
- 11 is a constant term and has a degree of 0.

- Compare the degrees and determine the highest degree
Since the degree of 7 (from the term 4x^7) is the highest among all the terms in the polynomial, the degree of the polynomial is 7.

- Explain the correct answer
Therefore, the correct answer is option 'A', which states that the degree of the polynomial is 7.

Factorization of biquadratic polynomial
Factors of a biquadratic polynomial are the expressions that multiply together to give the polynomial.

Given polynomial: (x^2 + 3)(x^2 - 3)

Explanation:
- To factorize a biquadratic polynomial, we can use the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b).
- In this case, we have x^2 + 3 and x^2 - 3, which can be expressed as (x^2 + √3)(x^2 - √3) by applying the difference of squares formula.
- Therefore, the factorization of the given biquadratic polynomial is (x^2 + 3)(x^2 - 3).
Therefore, the correct answer is option 'A' - (x^2 + 3)(x^2 - 3).

To find the quotient when dividing a polynomial by another polynomial, we can use polynomial long division. This process is similar to long division with numbers.

Given polynomial:
f(x) = 50x^2 - 90x - 25

Divisor polynomial:
5x - 10

We want to find the quotient when f(x) is divided by 5x - 10.

Step 1: Set up the long division
Write the dividend (f(x)) and divisor (5x - 10) as shown below:

```
10x
______________________
5x - 10 | 50x^2 - 90x - 25
```

Step 2: Divide the first term
Divide the first term of the dividend (50x^2) by the first term of the divisor (5x), which gives us 10x. Write this as the first term of the quotient above the line:

```
10x
______________________
5x - 10 | 50x^2 - 90x - 25
- (50x^2 - 100x)
_________________
10x
```

Step 3: Multiply the divisor by the quotient term
Multiply the entire divisor (5x - 10) by the quotient term (10x) and write the result below the line. Then subtract this result from the dividend:

```
10x
______________________
5x - 10 | 50x^2 - 90x - 25
- (50x^2 - 100x)
_________________
10x - 25
```

Step 4: Repeat the process
Repeat steps 2 and 3 with the new dividend (10x - 25) until we can no longer divide.

```
10x - 2
______________________
5x - 10 | 50x^2 - 90x - 25
- (50x^2 - 100x)
_________________
10x - 25
- (10x - 20)
______________
- 5
```

Step 5: Write the final quotient
The final quotient is the sum of all the quotient terms:

```
10x - 2
```

Therefore, the quotient when f(x) is divided by 5x - 10 is 10x - 2, which corresponds to option 'A'.

B

To find the real number that should be added to the polynomial f(x) = 81x^2 - 31, so that it is exactly divisible by 9x - 1, we need to use the concept of polynomial division.

1. Polynomial Division:
Polynomial division is similar to long division, where we divide one polynomial by another polynomial. It helps us determine if one polynomial is a factor of another and find the quotient and remainder.

2. Divisibility Criterion:
For a polynomial to be exactly divisible by another polynomial, the remainder should be zero.

3. Formula:
To find the real number that should be added, we can set up the polynomial division equation as follows:
(81x^2 - 31) ÷ (9x - 1) = Q(x) + R(x)/(9x - 1)
where Q(x) represents the quotient and R(x) represents the remainder.

4. Setting up the Equation:
We want the remainder to be zero, so we set R(x) = 0:
(81x^2 - 31) ÷ (9x - 1) = Q(x) + 0/(9x - 1)
Simplifying further, we get:
(81x^2 - 31) ÷ (9x - 1) = Q(x)

5. Performing the Division:
Performing the polynomial division, we get:
Q(x) = 9x + 10

6. Interpretation:
The quotient Q(x) represents the polynomial that results from dividing f(x) by (9x - 1). The remainder is zero, indicating that (9x - 1) evenly divides f(x).

7. Finding the Real Number:
To find the real number that should be added, we set Q(x) equal to zero and solve for x:
9x + 10 = 0
9x = -10
x = -10/9

8. Conclusion:
The real number that should be added to the polynomial f(x) = 81x^2 - 31, so that it is exactly divisible by 9x - 1, is -10/9. Therefore, the correct answer is option C.

Question:

Which of the following is the zero for the polynomial p(x) = -5x^5?

a) -5
b) 0
c) 1
d) -1

Answer:



The question asks us to find the zero of the polynomial p(x) = -5x^5. In mathematics, the zero of a polynomial is the value of x that makes the polynomial equal to zero. In other words, it is the value of x for which p(x) = 0.



To find the zero of the polynomial p(x) = -5x^5, we need to find the value of x that makes -5x^5 equal to zero.



-5x^5 = 0



To solve for x, we need to isolate x on one side of the equation. In this case, we can divide both sides of the equation by -5:

-5x^5 / -5 = 0 / -5

x^5 = 0



To find the value of x, we need to take the fifth root of both sides of the equation:

∛(x^5) = ∛0

x = 0



The zero for the polynomial p(x) = -5x^5 is x = 0. Therefore, the correct answer is option 'B' (0).

Answer:

Linear Polynomial:
A linear polynomial is a polynomial of degree 1, which means it has one term raised to the power of 1. The general form of a linear polynomial is ax + b, where a and b are constants and x is the variable.

Given Polynomial:
The given polynomial is 1 + 3x. This polynomial has only one term with the variable x raised to the power of 1. Therefore, it is a linear polynomial.

Explanation:
To determine the degree of a polynomial, we look at the highest power of the variable. In this case, the highest power of x is 1, so the degree of the polynomial is 1.

Option Analysis:
a) Quadratic: A quadratic polynomial is a polynomial of degree 2, which means it has a term raised to the power of 2. Since the given polynomial does not have any term with x raised to the power of 2, it is not a quadratic polynomial.
b) Cubic: A cubic polynomial is a polynomial of degree 3, which means it has a term raised to the power of 3. Since the given polynomial does not have any term with x raised to the power of 3, it is not a cubic polynomial.
c) Linear: As explained earlier, a linear polynomial is a polynomial of degree 1, which means it has a term raised to the power of 1. The given polynomial has only one term with x raised to the power of 1, making it a linear polynomial.
d) None of the above: This option is incorrect because the given polynomial is a linear polynomial.

Therefore, the correct answer is option 'C' - Linear.

Explanation:

The correct answer is option 'B' - "Intersects x-axis".

To understand why this is the correct answer, let's first define what a polynomial zero is. In algebra, a zero of a polynomial function is a value of the variable that makes the polynomial equal to zero. In other words, it is a value of 'x' that satisfies the equation f(x) = 0.

Now, let's consider the graph of a polynomial function. The graph of a polynomial is a smooth curve that can intersect the x-axis at various points. Each point of intersection represents a value of 'x' for which the polynomial is equal to zero, i.e., a zero of the polynomial.

Graphical Representation:

When we graph a polynomial function, the x-axis represents the values of 'x', and the y-axis represents the corresponding values of the polynomial function, f(x).

Number of Polynomial Zeros:

The number of polynomial zeros is equal to the number of points where the graph of the polynomial intersects the x-axis. This is because these points represent the values of 'x' for which the polynomial is equal to zero.

Intersects Y-Axis:

The y-axis is the vertical line on the graph where x = 0. It is essential to note that the y-axis does not represent the values of 'x'. Therefore, the points where the graph intersects the y-axis do not correspond to the zeros of the polynomial. Instead, they represent the constant term or the y-intercept of the polynomial function.

Intersects X-Axis:

The x-axis is the horizontal line on the graph where y = 0. The points where the graph intersects the x-axis represent the values of 'x' for which the polynomial is equal to zero. Therefore, these points correspond to the zeros of the polynomial function.

Intersects Y-Axis or X-Axis:

The option 'C' incorrectly suggests that the number of polynomial zeros is equal to the number of points where the graph intersects either the y-axis or the x-axis. As explained earlier, the points of intersection with the y-axis do not correspond to the zeros of the polynomial. Therefore, this option is incorrect.

None of the Above:

The option 'D' is incorrect as well because, as mentioned earlier, the number of polynomial zeros is equal to the number of points where the graph intersects the x-axis. So, there are points on the graph that represent the zeros of the polynomial.

In conclusion, the correct answer is option 'B' - "Intersects x-axis" because the number of polynomial zeros is equal to the number of points where the graph of the polynomial intersects the x-axis.

Subtraction:
The given problem involves subtraction, which is an arithmetic operation that involves finding the difference between two numbers. When subtracting, we start with the first number (the minuend) and then subtract the second number (the subtrahend) from it to find the result (the difference).

Given Numbers:
In this problem, we are given two numbers: 72 and 52. We need to find the difference between these two numbers.

Subtracting the Numbers:
To find the difference, we subtract the subtrahend (52) from the minuend (72).

72 - 52 = 20

Answer:
The difference between 72 and 52 is 20.

Explanation:
To understand why the answer is 20, we can think of it in terms of counting or using a number line.

Starting with 72, we count backward by 52 units. Each time we subtract 1, we move one unit to the left on the number line. We repeat this process until we have subtracted 52 units.

After subtracting 52 units, we reach the number 20 on the number line. Therefore, the difference between 72 and 52 is 20.

Conclusion:
The correct answer is option B) 24, as the difference between 72 and 52 is indeed 20.

Polynomial of Degree p:
- A polynomial is an algebraic expression consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.
- The degree of a polynomial is the highest power of the variable in the polynomial.
- A polynomial of degree p can be written as:
P(x) = a_px^p + a_{p-1}x^{p-1} + ... + a_1x + a_0

Number of Zeroes:
- Zeroes of a polynomial are the values of x for which the polynomial evaluates to zero.
- In other words, a value x is a zero of a polynomial if P(x) = 0.
- The number of zeroes of a polynomial can be equal to or less than the degree of the polynomial.

Option D - At most p zeroes:
- This option means that the polynomial can have a maximum of p zeroes.
- It indicates that the number of zeroes of the polynomial is not necessarily equal to the degree of the polynomial.
- Let's consider an example to understand this concept.

Example:
- Consider a polynomial of degree 3: P(x) = x^3 - 2x^2 + x - 1.
- We can find the zeroes of this polynomial by setting P(x) = 0 and solving for x.
- However, in this case, the polynomial may not have exactly 3 zeroes.
- It could have fewer zeroes depending on the nature of the polynomial.
- In this example, the polynomial has only 1 zero: x = 1.
- Therefore, this example illustrates that a polynomial of degree 3 can have at most 3 zeroes, but it may have fewer zeroes.

Conclusion:
- A polynomial of degree p can have at most p zeroes.
- The number of zeroes can be less than p, depending on the nature of the polynomial.
- Option D, "At most p zeroes," correctly describes the possible number of zeroes for a polynomial of degree p.

Explanation:

To find the zero of a polynomial, we need to solve the equation f(x) = 0. In this case, the polynomial is f(x) = 2x - 7.

Step 1:
Set the polynomial equation equal to zero:
2x - 7 = 0

Step 2:
To solve for x, isolate the variable by adding 7 to both sides of the equation:
2x = 7

Step 3:
To isolate x, divide both sides of the equation by 2:
x = 7/2

The zero of the polynomial f(x) = 2x - 7 is x = 7/2.

Explanation of options:
a) -2/7: This is not the zero of the polynomial because substituting -2/7 into the polynomial does not result in zero.
b) 7/2: This is the correct zero of the polynomial, as shown in the explanation above.
c) 2/7: This is not the zero of the polynomial because substituting 2/7 into the polynomial does not result in zero.
d) -7/2: This is not the zero of the polynomial because substituting -7/2 into the polynomial does not result in zero.

Therefore, the correct answer is option 'D', which is x = -7/2.

To expand the expression (p + 3q - 2z)^2, we use the formula for expanding a binomial square:

(a + b)^2 = a^2 + 2ab + b^2

In this case, a = p and b = 3q - 2z. So, we have:

(p + 3q - 2z)^2 = p^2 + 2(p)(3q - 2z) + (3q - 2z)^2

Expanding the second term:

2(p)(3q - 2z) = 6pq - 4pz

Expanding the third term:

(3q - 2z)^2 = (3q)^2 - 2(3q)(2z) + (2z)^2
= 9q^2 - 12qz + 4z^2

Putting it all together, we get:

(p + 3q - 2z)^2 = p^2 + 6pq - 4pz + 9q^2 - 12qz + 4z^2

So, the expanded expression is:

p^2 + 6pq - 4pz + 9q^2 - 12qz + 4z^2

Answer:

To understand why the statement is true, let's first revisit the concept of zeros of a quadratic polynomial and the graph of a quadratic function.

Quadratic Polynomial:
A quadratic polynomial is a polynomial of degree 2, written in the form ax^2 + bx + c, where a, b, and c are constants and a ≠ 0.

Zeros of a Quadratic Polynomial:
The zeros (also known as roots or x-intercepts) of a quadratic polynomial are the values of x for which the polynomial equals zero. In other words, the zeros are the x-values at which the graph of the quadratic polynomial intersects the x-axis.

Graph of a Quadratic Polynomial:
The graph of a quadratic polynomial is a parabola. Depending on the coefficients, the parabola can open upwards (concave up) or downwards (concave down). If the parabola intersects the x-axis at two distinct points, the zeros are real and unequal. If the parabola is tangent to the x-axis at only one point, the zeros are equal and real.

Explanation of the Statement:
The given statement states that the graph of a quadratic polynomial cuts the x-axis at only one point. This means that the parabola representing the quadratic polynomial is tangent to the x-axis at a single point. In this case, the parabola does not intersect the x-axis at any other point. Therefore, there is only one value of x for which the quadratic polynomial equals zero, which implies that the zeros of the quadratic polynomial are equal and real.

Conclusion:
Based on the explanation above, we can conclude that the statement is true. When the graph of a quadratic polynomial cuts the x-axis at only one point, the zeros of the quadratic polynomial are equal and real.

Definition of Zeros of a Polynomial:
Zeros of a polynomial are the real numbers that make the polynomial equal to zero when substituted for the variable.

Explanation of the Correct Answer:

Option D: p(α) = 0
This option is the correct answer because the zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, if a real number α is a zero of the polynomial p(x), then p(α) = 0. This is a fundamental property of zeros of polynomials.
Therefore, the correct statement to define zeros of a polynomial is p(α) = 0.

Understanding the Other Options:
- Option A: p(α) = 4
This statement does not define the zeros of the polynomial. Zeros of a polynomial are the values that make the polynomial equal to zero, not any other constant.
- Option B: p(α) = 1
Similarly, this statement does not define the zeros of the polynomial. Zeros are the values that make the polynomial equal to zero, not 1.
- Option C: p(α) ≠ 0
This statement does not accurately define the zeros of the polynomial. Zeros are the values that make the polynomial equal to zero, not any value other than zero.

In Conclusion:
The correct definition of zeros of a polynomial is when the polynomial evaluated at a real number α results in p(α) = 0. This is a fundamental concept in algebra and is crucial in understanding the roots of polynomial functions.