Free Pre-Calculus Polynomial Functions Questions and Answers — Questions and Answers

Question 1: What is the degree of P(x)=3x^4-7x^2-2x^7-x+4 polynomial function?

1
2
4
7 (Correct answer)

Correct answer: 7

The degree of a polynomial is determined by the highest exponent of the variable in the polynomial expression. In the polynomial P(x) = 3x^4 - 7x^2 - 2x^7 - x + 4, the exponents of x are 4, 2, 7, 1, and 0 (for the constant term). The largest exponent among these is 7, so the degree of the polynomial is 7.

Question 2: The equation f(x) = 5x^4-8x^3+4x^2-6x+3 has how many roots or zeros?

0
1
5
4 (Correct answer)

Correct answer: 4

According to the Fundamental Theorem of Algebra, a polynomial function of degree 'n' will have exactly 'n' roots or zeros in the complex number system, counting multiplicity. The given polynomial f(x) = 5x^4 - 8x^3 + 4x^2 - 6x + 3 has a highest exponent of 4. Therefore, its degree is 4, meaning it has exactly 4 roots or zeros.

Question 3: P(x) = x^3+6x^2+9x+54 has a polynomial function; find all its zeros.

-6, 3i, -3i (Correct answer)
6, 3i, -3i
-6, 3, -3
6, 3, -3

Correct answer: -6, 3i, -3i

To find the zeros of P(x) = x^3 + 6x^2 + 9x + 54, we can factor by grouping. Grouping the terms gives x²(x+6) + 9(x+6), which factors further into (x²+9)(x+6). Setting each factor to zero, x+6=0 yields x=-6, and x²+9=0 yields x²=-9, so x=±√(-9)=±3i. Thus, the zeros are -6, 3i, and -3i.

Question 4: What is another term for the point where a function intersects the x-axis?

x-intercept
Zero
Root
All of these (Correct answer)

Correct answer: All of these

The point where a function intersects the x-axis is characterized by the y-coordinate being zero. This point is commonly referred to as an x-intercept. It is also called a 'zero' of the function because f(x) equals zero at that point, and a 'root' of the equation f(x)=0. Therefore, all these terms correctly describe the same concept.

Question 5: The polynomial can be represented as...

Odd degree, negative leading coefficient.
Odd degree, positive leading coefficient.
Even degree, negative leading coefficient.
Even degree, positive leading coefficient. (Correct answer)

Correct answer: Even degree, positive leading coefficient.

The end behavior of a polynomial graph is determined by its degree and the sign of its leading coefficient. A polynomial with an even degree and a positive leading coefficient will have both ends of its graph pointing upwards. This means as x approaches positive or negative infinity, y also approaches positive infinity.

Question 6: What degree is shown in the picture?

Odd (Correct answer)
Even
Positive
Negative

Correct answer: Odd

The end behavior of a polynomial graph indicates its degree. If the ends of the graph point in opposite directions (one end up and the other end down), the polynomial has an odd degree. This characteristic holds true regardless of whether the leading coefficient is positive or negative.

Question 7: Classify based on the Degree: 4x^3 - 5x^2 + 2x - 1

2
10
4
3 (Correct answer)

Correct answer: 3

The degree of a polynomial is the highest exponent of the variable in the expression. For the polynomial 4x^3 - 5x^2 + 2x - 1, the exponents of x are 3, 2, 1, and 0 (for the constant term). The largest exponent is 3, so the polynomial is classified as having a degree of 3.

Question 8: f(x) = 2x^3 - 3x^2 + 4x -10 How many degrees does f(x) have?

1
2
0
3 (Correct answer)

Correct answer: 3

The degree of a polynomial function is determined by the highest exponent of the variable present in the function. In f(x) = 2x^3 - 3x^2 + 4x - 10, the terms have exponents 3, 2, 1, and 0 respectively. The highest exponent among these is 3, making the degree of f(x) equal to 3.

Question 9: f(x) = -3x^4 + x^3 - 2x^2 + x - 1 How many degrees does f(x) have?

4 (Correct answer)
3
2
1

Correct answer: 4

The degree of a polynomial is defined as the highest exponent of the variable in the polynomial expression. For the function f(x) = -3x^4 + x^3 - 2x^2 + x - 1, the exponents of x are 4, 3, 2, and 1. The largest of these exponents is 4, so the degree of f(x) is 4.

Question 10: Given: f(x) = -3x^4 + x^3 - 2x^2 + x - 1, What is the y-intercept of f(x)?

(0, -3)
(-3, 0)
(-1, 0)
(0, -1) (Correct answer)

Correct answer: (0, -1)

The y-intercept of a function is the point where the graph crosses the y-axis, which occurs when x=0. To find it, substitute x=0 into the function f(x) = -3x^4 + x^3 - 2x^2 + x - 1. This yields f(0) = -3(0)^4 + (0)^3 - 2(0)^2 + (0) - 1 = -1. Therefore, the y-intercept is (0, -1).

Question 11: Determine the zeros of f(x) if f(x) = x^3 - 3x^2 - 4x.

{-1, 4}
{-4, 0, 1}
{-4, 1}
{-1, 0, 4} (Correct answer)

Correct answer: {-1, 0, 4}

To find the zeros of f(x) = x^3 - 3x^2 - 4x, set the function equal to zero and factor the polynomial. First, factor out the common term 'x' to get x(x^2 - 3x - 4) = 0. Then, factor the quadratic expression (x^2 - 3x - 4) into (x-4)(x+1). Setting each factor to zero gives the zeros: x=0, x-4=0 (so x=4), and x+1=0 (so x=-1). Thus, the zeros are {-1, 0, 4}.

Question 12: f(x) = x3 - 9x^2 - 45x - 27 : Determine the remaining zeros if -3 is a zero of f(x).

{-3, -6 ± 3√5}
(-3, 3, 9}
{-3, 0, 3}
{-3, 6 ± 3√5} (Correct answer)

Correct answer: {-3, 6 ± 3√5}

Since -3 is a zero of f(x) = x^3 - 9x^2 - 45x - 27, (x+3) is a factor. Using synthetic division with -3, we divide the polynomial to get the quotient x^2 - 12x - 9. To find the remaining zeros, set this quadratic equal to zero and use the quadratic formula: x = [12 ± √((-12)^2 - 4(1)(-9))] / 2(1) = [12 ± √(144 + 36)] / 2 = [12 ± √180] / 2. Simplifying √180 to 6√5, we get x = [12 ± 6√5] / 2 = 6 ± 3√5. Therefore, the remaining zeros are 6 + 3√5 and 6 - 3√5.

Question 13: Determine whether (x-3) is a factor of (3x^3+10x^2-x-12)

Yes, it is a factor
No, it is not a factor (Correct answer)
Maybe
None of the above

Correct answer: No, it is not a factor

According to the Factor Theorem, (x-c) is a factor of a polynomial P(x) if and only if P(c) = 0. Here, we test if (x-3) is a factor, so c=3. Substitute x=3 into the polynomial P(x) = 3x^3 + 10x^2 - x - 12: P(3) = 3(3)^3 + 10(3)^2 - (3) - 12 = 3(27) + 10(9) - 3 - 12 = 81 + 90 - 3 - 12 = 171 - 15 = 156. Since P(3) = 156 and not 0, (x-3) is not a factor of the polynomial.

Question 14: Determine the roots of the following: Solve (n-2)(4n+3)=0

(-4/3, 2)
(4/3, -2)
(3/4, -2)
(-3/4, 2) (Correct answer)

Correct answer: (-3/4, 2)

To determine the roots of the equation (n-2)(4n+3)=0, we set each factor equal to zero. For the first factor, n-2=0, which gives n=2. For the second factor, 4n+3=0, which means 4n=-3, so n=-3/4. Therefore, the roots of the equation are -3/4 and 2.

Question 15: Determine all the factors of x^3 -3x^2 -4x +12 if -2 is a zero.

(x-2) (x-2) (x+3)
(x-2) (x+2) (x+3)
(x-2) (x+2) (x-3) (Correct answer)
(x+2) (x+2) (x+3)

Correct answer: (x-2) (x+2) (x-3)

Since -2 is a zero of x^3 - 3x^2 - 4x + 12, (x+2) is a factor. Using synthetic division with -2, we divide the polynomial to get the quotient x^2 - 5x + 6. This quadratic expression can be factored further into (x-2)(x-3). Therefore, all the factors of the original polynomial are (x+2), (x-2), and (x-3).

Question 16: The zeros of a polynomial function with rational coefficients are 3 + 5 and i. Find all additional zeros.

3 + √5
−3 - √5 and -i
−3 - √5 and i (Correct answer)
−3 - √5

Correct answer: −3 - √5 and i

The Conjugate Root Theorem states that if a polynomial has rational coefficients, then irrational roots involving square roots and complex roots must occur in conjugate pairs. If the given zeros were -3 + √5 and -i, then their conjugates would also be zeros. The conjugate of -3 + √5 is -3 - √5, and the conjugate of -i is i. Thus, the additional zeros would be -3 - √5 and i.

Free Pre-Calculus Polynomial Functions Questions and Answers

What is the degree of P(x)=3x^4-7x^2-2x^7-x+4 polynomial function?