2.1 MCQs-Polynomials

Polynomials MCQs

Basic Concepts

1. Which of the following is NOT a polynomial?

1. $5x^3 - 2x + 1$

2. $\sqrt{x} + 3$

3. $4x^2 + \frac{1}{2}x - 7$

4. $-9$

Show me the answer

Answer: 2. $\sqrt{x} + 3$

Explanation:

• A polynomial is an expression consisting of variables and coefficients, involving only non-negative integer exponents.

• $\sqrt{x} = x^{1/2}$ has a fractional exponent (1/2), which is not a non-negative integer.

• Option 1 is a cubic polynomial.

• Option 3 is a quadratic polynomial.

• Option 4 is a constant polynomial.

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2. The degree of the polynomial $4x^5 - 2x^3 + 7x - 10$ is:

1. 3

2. 4

3. 5

4. 10

Show me the answer

Answer: 3. 5

Explanation:

• The degree of a polynomial is the highest power of the variable x with a non-zero coefficient.

• In $4x^5 - 2x^3 + 7x - 10$, the terms have exponents 5, 3, 1, and 0.

• The highest exponent is 5, so the degree is 5.

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3. A polynomial of degree 2 is called:

1. Linear

2. Quadratic

3. Cubic

4. Quartic

Show me the answer

Answer: 2. Quadratic

Explanation:

• Polynomials are classified by their degree:

  • Degree 0: Constant

  • Degree 1: Linear (e.g., $2x+1$)

  • Degree 2: Quadratic (e.g., $x^2 - 5x + 6$)

  • Degree 3: Cubic

  • Degree 4: Quartic

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Operations on Polynomials

4. The sum of the polynomials $(3x^2 + 2x - 5)$ and $(x^2 - 4x + 7)$ is:

1. $4x^2 - 2x + 2$

2. $3x^2 - 2x + 2$

3. $4x^2 + 6x + 2$

4. $2x^2 - 2x + 12$

Show me the answer

Answer: 1. $4x^2 - 2x + 2$

Explanation:

• Add like terms (terms with the same power of x):

  • $x^2$ terms: $3x^2 + x^2 = 4x^2$

  • $x$ terms: $2x + (-4x) = -2x$

  • Constant terms: $-5 + 7 = 2$

• Therefore, the sum is $4x^2 - 2x + 2$.

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5. The product $(x + 3)(x - 2)$ equals:

1. $x^2 + x - 6$

2. $x^2 + 5x - 6$

3. $x^2 - x - 6$

4. $x^2 + x + 6$

Show me the answer

Answer: 1. $x^2 + x - 6$

Explanation:

• Use the distributive property (FOIL method):

  • First: $x \cdot x = x^2$

  • Outer: $x \cdot (-2) = -2x$

  • Inner: $3 \cdot x = 3x$

  • Last: $3 \cdot (-2) = -6$

• Combine like terms: $x^2 + (-2x + 3x) - 6 = x^2 + x - 6$

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6. When dividing $(2x^3 - 5x^2 + 3x - 7)$ by $(x - 2)$ using synthetic division, the first number brought down is:

1. 2

2. -5

3. 3

4. -7

Show me the answer

Answer: 1. 2

Explanation:

• In synthetic division, you use the coefficients of the dividend polynomial: 2, -5, 3, -7 (for $2x^3 - 5x^2 + 3x - 7$).

• The first step is to "bring down" the leading coefficient (the coefficient of the highest degree term), which is 2.

• The divisor is $(x - 2)$, so you use the zero of the divisor, which is 2.

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Factorization and Roots

7. One factor of the quadratic $x^2 - 5x + 6$ is:

1. $(x + 2)$

2. $(x - 2)$

3. $(x + 3)$

4. $(x - 6)$

Show me the answer

Answer: 2. $(x - 2)$

Explanation:

• Factor the quadratic: We look for two numbers that multiply to 6 (constant term) and add to -5 (coefficient of x).

• Those numbers are -2 and -3.

• Therefore, $x^2 - 5x + 6 = (x - 2)(x - 3)$.

• So, $(x - 2)$ is a factor.

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8. The roots (or zeros) of the polynomial $x^2 - 9$ are:

1. $x = 3$ only

2. $x = -3$ only

3. $x = 3$ and $x = -3$

4. $x = 0$ only

Show me the answer

Answer: 3. $x = 3$ and $x = -3$

Explanation:

• Set the polynomial equal to zero: $x^2 - 9 = 0$.

• This is a difference of squares: $(x - 3)(x + 3) = 0$.

• By the zero-product property: $x - 3 = 0$ gives $x = 3$, and $x + 3 = 0$ gives $x = -3$.

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9. If $(x - 1)$ is a factor of the polynomial $P(x)$, then according to the Factor Theorem:

1. $P(0) = 1$

2. $P(1) = 0$

3. $P(-1) = 0$

4. $P(0) = -1$

Show me the answer

Answer: 2. $P(1) = 0$

Explanation:

• The Factor Theorem states: $(x - c)$ is a factor of polynomial $P(x)$ if and only if $P(c) = 0$.

• Therefore, if $(x - 1)$ is a factor, then plugging $x = 1$ into $P(x)$ must yield zero: $P(1) = 0$.

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The Remainder Theorem

10. According to the Remainder Theorem, when a polynomial $P(x)$ is divided by $(x - c)$, the remainder is:

1. $P(0)$

2. $P(c)$

3. $P(-c)$

4. $P(1)$

Show me the answer

Answer: 2. $P(c)$

Explanation:

• The Remainder Theorem states: If a polynomial $P(x)$ is divided by $(x - c)$, the remainder is the constant $P(c)$.

• This provides a quick way to evaluate the remainder without performing long division.

• Example: The remainder of $P(x)$ divided by $(x - 5)$ is $P(5)$.

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11. When $P(x) = 2x^3 - 4x^2 + 3x - 5$ is divided by $(x - 2)$, the remainder is:

1. 1

2. -1

3. 5

4. -5

Show me the answer

Answer: 1. 1

Explanation:

• By the Remainder Theorem, the remainder is $P(2)$.

• Calculate $P(2)$: $2(2)^3 - 4(2)^2 + 3(2) - 5$ $= 2(8) - 4(4) + 6 - 5$ $= 16 - 16 + 6 - 5 = 1$

• So the remainder is 1.

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Graphs and Behavior

12. The graph of an even-degree polynomial with a positive leading coefficient:

1. Rises to the left and falls to the right

2. Falls to the left and rises to the right

3. Rises to both the left and right

4. Falls to both the left and right

Show me the answer

Answer: 3. Rises to both the left and right

Explanation:

• The "end behavior" of a polynomial is determined by its degree and leading coefficient.

• For an even degree:

  • Positive leading coefficient: Graph rises as $x \to \infty$ and as $x \to -\infty$.

  • Negative leading coefficient: Graph falls as $x \to \infty$ and as $x \to -\infty$.

• For an odd degree, the ends go in opposite directions.

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13. A polynomial of degree n can have at most how many real roots (x-intercepts)?

1. n

2. n-1

3. n+1

4. n/2

Show me the answer

Answer: 1. n

Explanation:

• The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n complex roots (counting multiplicity).

• Therefore, the number of real roots is at most n.

• Example: A cubic polynomial (degree 3) can have 1, 2, or 3 real roots.

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14. If a polynomial has a root at $x = a$ with multiplicity 2, what does this imply about its graph at that point?

1. The graph crosses the x-axis at $(a, 0)$.

2. The graph touches the x-axis at $(a, 0)$ but does not cross it.

3. The graph has a vertical asymptote at $x = a$.

4. The graph has a hole at $x = a$.

Show me the answer

Answer: 2. The graph touches the x-axis at $(a, 0)$ but does not cross it.

Explanation:

• Multiplicity refers to the number of times a particular root is repeated.

• If the multiplicity is odd, the graph crosses the x-axis at that root.

• If the multiplicity is even, the graph touches the x-axis at that root (is tangent to it) but does not cross it.

• A root of multiplicity 2 is an even number, so the graph touches and turns around at that point.

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Synthetic Division and Applications

15. Using synthetic division to divide $P(x) = x^3 - 6x^2 + 11x - 6$ by $(x - 1)$ yields a quotient of:

1. $x^2 - 5x + 6$

2. $x^2 - 5x - 6$

3. $x^2 + 5x + 6$

4. $x^2 - 5x + 10$

Show me the answer

Answer: 1. $x^2 - 5x + 6$

Explanation:

• Set up synthetic division with the zero of the divisor (c = 1) and coefficients [1, -6, 11, -6].

• Process:

  • Bring down 1.

  • Multiply 1 * 1 = 1, add to -6 to get -5.

  • Multiply 1 * -5 = -5, add to 11 to get 6.

  • Multiply 1 * 6 = 6, add to -6 to get 0 (confirming remainder 0).

• The numbers brought down (1, -5, 6) are the coefficients of the quotient, starting one degree lower.

• So the quotient is $1x^2 - 5x + 6$, which is $x^2 - 5x + 6$.