**POLYNOMIALS – COMPREHENSIVE CHEAT SHEET**
**1. POLYNOMIAL BASICS**
• **Polynomial Definition**: An expression in one or more variables with non-negative integer exponents and real coefficients
• **Degree**: The highest power of the variable in a polynomial
• **Non-polynomials**: Expressions like 1/(x+1), √x, x⁻², (2x³ + 3x – 1)/(x² + 2) are NOT polynomials (contain negative or fractional exponents, or variables in denominator)
**Linear Polynomial**: Degree 1 → Form: ax + b, where a ≠ 0
**Quadratic Polynomial**: Degree 2 → Form: ax² + bx + c, where a ≠ 0
**Cubic Polynomial**: Degree 3 → Form: ax³ + bx² + cx + d, where a ≠ 0
**2. VALUE OF A POLYNOMIAL**
• **Definition**: If p(x) is a polynomial and k is any real number, then p(k) is the value obtained by substituting x = k in p(x)
• **Method**: Replace x with k and calculate
**3. ZEROES OF A POLYNOMIAL (ROOTS)**
• **Definition**: A real number k is a zero of polynomial p(x) if p(k) = 0
• **Zero of Linear Polynomial ax + b**: k = –b/a
• **Quadratic/Cubic Polynomials**: Can have multiple zeroes (found by solving p(x) = 0)
• **Important**: A polynomial of degree n can have AT MOST n zeroes
**4. GEOMETRICAL MEANING OF ZEROES**
**For Linear Polynomial y = ax + b (a ≠ 0)**:
• Graph is a straight line
• Intersects x-axis at exactly ONE point: (–b/a, 0)
• The zero is the x-coordinate of this intersection point
• Example: y = 2x + 3 intersects x-axis at (–3/2, 0)
**For Quadratic Polynomial y = ax² + bx + c (a ≠ 0)**:
• Graph is a parabola (U-shaped or ∩-shaped)
• Opens upward if a > 0; opens downward if a < 0
• Zeroes are x-coordinates where parabola meets x-axis
**Three Cases**:
**For Cubic Polynomial y = ax³ + bx² + cx + d (a ≠ 0)**:
• Graph is a cubic curve
• Can have 1, 2, or 3 real zeroes (x-coordinates of intersection points with x-axis)
• Example: y = x³ – 4x intersects x-axis at x = –2, 0, 2 (three zeroes)
**5. RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS**
**For Quadratic Polynomial ax² + bx + c with zeroes α and β**:
• **Sum of zeroes**: α + β = –b/a
• **Product of zeroes**: αβ = c/a
• **Verification Formula**: If α and β are zeroes, then:
**For Cubic Polynomial ax³ + bx² + cx + d with zeroes α, β, and γ**:
• **Sum of zeroes**: α + β + γ = –b/a
• **Sum of products (taken two at a time)**: αβ + βγ + γα = c/a
• **Product of all zeroes**: αβγ = –d/a
**6. DIVISION ALGORITHM FOR POLYNOMIALS**
• **Statement**: For any two polynomials p(x) and g(x) where g(x) ≠ 0, there exist unique polynomials q(x) and r(x) such that:
• **q(x)** = quotient; **r(x)** = remainder
• If r(x) = 0, then g(x) is a factor of p(x)
• **Application**: Finding remainders without full division; factorization
**7. REMAINDER THEOREM**
• **Statement**: If a polynomial p(x) is divided by (x – a), then the remainder is p(a)
• **Proof Insight**: By division algorithm, p(x) = (x – a) · q(x) + r where r is constant. Substituting x = a gives p(a) = 0 + r, so r = p(a)
• **Usage**: To find remainder when p(x) is divided by (x – a), simply calculate p(a)
• **Example**: Remainder when x² – 3x – 4 is divided by (x – 2) is p(2) = 4 – 6 – 4 = –6
**8. FACTOR THEOREM**
• **Statement**: (x – a) is a factor of polynomial p(x) if and only if p(a) = 0
• **Proof Insight**: By remainder theorem, if p(a) = 0, then remainder is 0 → p(x) = (x – a) · q(x), so (x – a) is a factor
• **Converse**: If (x – a) is a factor, then remainder = 0, so p(a) = 0
• **Usage**: To check if (x – a) divides p(x), calculate p(a). If p(a) = 0, then (x – a) is a factor
• **Finding Zeroes**: If p(a) = 0, then 'a' is a zero; equivalently (x – a) is a factor
**9. FACTORIZATION OF POLYNOMIALS**
**Quadratic Polynomial ax² + bx + c**:
• If zeroes are α and β: ax² + bx + c = a(x – α)(x – β)
• **Method 1** (Completing the square): Write as perfect square ± constant
• **Method 2** (Splitting middle term): Find two numbers whose product = ac and sum = b
• **Method 3** (Quadratic formula): Zeroes are [–b ± √(b² – 4ac)]/(2a)
**Cubic Polynomial ax³ + bx² + cx + d**:
• If one zero is known, use factor theorem to extract factor (x – a)
• Divide p(x) by (x – a) to get a quadratic; then factor the quadratic
• **Method**: If p(a) = 0, then p(x) = (x – a) · q(x) where q(x) is quadratic
**10. IMPORTANT RESULTS**
• **Polynomial with given zeroes**: If zeroes are α and β, polynomial is k(x – α)(x – β) for any constant k
• **Number of zeroes**: A polynomial of degree n has AT MOST n zeroes
• **Real coefficients & Complex zeroes**: Complex zeroes occur in conjugate pairs for polynomials with real coefficients
• **Identical polynomials**: Two polynomials are identical if all corresponding coefficients are equal
**11. COMMON MISTAKES & HOW TO AVOID**
❌ **Mistake 1**: Confusing 'value at k' with 'zero' → Remember: zero means p(k) = 0; value can be any number
❌ **Mistake 2**: Forgetting a ≠ 0 in definitions → In ax² + bx + c, if a = 0, it's linear, not quadratic
❌ **Mistake 3**: Wrong signs in relationships → For ax² + bx + c: sum = –b/a (negative sign); product = c/a
❌ **Mistake 4**: Misapplying factor theorem → (x – a) is factor only if p(a) = 0, not p(–a) = 0
❌ **Mistake 5**: Remainder theorem error → Remainder when dividing by (x – a) is p(a), NOT p(–a)
❌ **Mistake 6**: Ignoring degree conditions in division → Always check: degree(remainder) < degree(divisor)
❌ **Mistake 7**: Assuming all quadratics factor over reals → Only if discriminant b² – 4ac ≥ 0
**12. PROBLEM-SOLVING STRATEGIES**
• **Finding zeroes**: Set p(x) = 0 and solve; or use factor theorem if one zero is given
• **Finding remainder**: Use remainder theorem (calculate p(a) directly)
• **Checking if (x – a) is factor**: Calculate p(a); if 0, then it's a factor
• **Complete factorization**: Extract known factors, reduce degree, repeat
• **Using zeroes-coefficient relationships**: Set up equations using sum and product formulas
Q1. A student observes that the graph of y = 2x + 3 crosses the x-axis at x = −3/2. Which statement best explains why this x-value is called a zero of the polynomial 2x + 3?
Answer: A — A zero is defined as a value of x where p(x) = 0, not merely where the graph intersects the x-axis; option B describes the geometric consequence but not the algebraic definition, which is what 'zero' means.
Q2. Assertion (A): A quadratic polynomial ax² + bx + c with a > 0 can have no real zeroes. Reason (R): When the parabola y = ax² + bx + c opens upward and does not intersect the x-axis, the polynomial has no real zeroes. Choose the correct option:
Answer: A — Both statements are true; R directly explains why A is true because the geometric fact (no x-intercept) is the reason the algebraic fact (no real zeroes) holds.
Q3. A linear polynomial p(x) = ax + b has exactly one zero at x = k. Which of the following must be true?
Answer: B — If k is a zero, then p(k) = 0, which means the point (k, 0) satisfies the equation y = ax + b; a horizontal line would have infinitely many zeroes or none, and b and a have no fixed sign restrictions.
Q4. Consider two quadratic polynomials: p(x) = x² − 5x + 6 and q(x) = −x² + 5x − 6. A student claims that p(x) and q(x) have the same zeroes because one is the negative of the other. Is this claim always true, sometimes true, or never true?
Answer: D — If p(k) = 0, then −p(k) = 0 as well, so q(k) = −p(k) = 0; the zeroes are identical, not opposite (verify: p(x) = 0 at x = 2, 3; q(x) = 0 also at x = 2, 3).
Q5. Assertion (A): A cubic polynomial ax³ + bx² + cx + d always has at least one real zero. Reason (R): A cubic polynomial has degree 3, which is odd, so its graph must cross the x-axis at least once. Choose the correct option:
Answer: A — Both are true; the odd degree ensures the polynomial takes both positive and negative values, so by continuity it must cross the x-axis, which is exactly why it has at least one real zero.
Q6. A quadratic polynomial has zeroes at x = 2 and x = 5. Which statement correctly relates the zeroes to the polynomial's properties?
Answer: A — Any quadratic with zeroes 2 and 5 can be written as a(x − 2)(x − 5), where a ≠ 0 is arbitrary; option B fixes a = 1, option C requires a > 0, and option D requires 10a > 0, all of which are unnecessary restrictions.
Q7. Assertion (A): If p(x) is a polynomial and p(k) = 0, then (x − k) is a factor of p(x). Reason (R): The zero of a linear polynomial ax + b is always x = −b/a. Choose the correct option:
Answer: B — Both are true statements, but R is about linear polynomials only and does not explain A, which is a general statement about any polynomial; A is known as the Factor Theorem.
Q8. A student graphs y = x² − 4 and observes it crosses the x-axis at x = −2 and x = 2. Based on this observation, which algebraic fact does the student verify?
Answer: A — The x-intercepts directly verify that p(−2) = 0 and p(2) = 0, which is the definition of zeroes; while B, C, and D are related properties, they are not directly verified by observing where the graph crosses the x-axis.
Q9. Assertion (A): Every polynomial of degree n has at most n zeroes. Reason (R): A polynomial can be written as a product of linear factors corresponding to its zeroes. Choose the correct option:
Answer: B — Both statements are true, but R describes factorization while A is a statement about the maximum count of zeroes; R is related but does not fully explain why the maximum is exactly n (the explanation requires the Fundamental Theorem of Algebra).
Q10. A polynomial p(x) has the property that p(0) = 5. Based solely on this information, which conclusion is valid?
Answer: B — p(0) gives the y-intercept, which equals the constant term; p(0) = 5 tells us nothing about whether 5 is a zero (it is not, since p(5) is unknown), the degree, or the existence of other zeroes.
What is the definition of a zero of polynomial p(x)?
A real number k is a zero of p(x) if p(k) = 0.
What is the general form of a quadratic polynomial?
ax² + bx + c, where a, b, c are real numbers and a ≠ 0.
How many zeroes can a quadratic polynomial have at most?
A quadratic polynomial can have at most 2 zeroes.
What does a zero of a polynomial represent on its graph?
A zero of p(x) is the x-coordinate of the point where the graph intersects the x-axis.
If p(x) = 2x + 3, what is its zero?
The zero is -3/2, found by solving 2x + 3 = 0.
What is the shape of the graph of a quadratic polynomial ax² + bx + c when a > 0?
The parabola opens upwards.
What is the difference between a cubic polynomial and a quadratic polynomial?
A cubic polynomial has degree 3, while a quadratic has degree 2; a cubic can have up to 3 zeroes.
When does a quadratic polynomial have exactly one zero?
When the parabola touches the x-axis at exactly one point (two coincident zeroes).
For linear polynomial ax + b (a ≠ 0), what is the zero?
The zero is -b/a, which equals -(Constant term)/(Coefficient of x).
Can a quadratic polynomial have no real zeroes? If yes, what does this mean graphically?
Yes; the parabola is completely above or below the x-axis and never intersects it.
Define the zero of a polynomial. Is –2 a zero of the polynomial p(x) = 3x² + 5x – 2? Justify your answer. [2 marks]
Zero is defined as k where p(k) = 0. Calculate p(–2) = 3(4) + 5(–2) – 2 and check if it equals zero.
Find the zeroes of the quadratic polynomial x² – 2x – 8 and verify that they are the x-coordinates of the points where the graph intersects the x-axis. [3 marks]
Factor x² – 2x – 8 = (x – 4)(x + 2) to find zeroes 4 and –2. Verify by substituting: p(4) = 0 and p(–2) = 0; these x-values are where the parabola touches the x-axis.
Explain with the help of a graph why a quadratic polynomial can have 0, 1, or 2 zeroes. How does the position of the parabola relative to the x-axis determine the number of zeroes? Also, state how many zeroes a cubic polynomial can have at most. [5 marks]
Describe three cases: (1) parabola cuts x-axis at two points → 2 distinct zeroes; (2) parabola touches x-axis at one point → 1 zero (repeated); (3) parabola lies completely above/below x-axis → no zeroes. Explain that 'a' (coefficient of x²) determines if parabola opens up/down. A cubic polynomial of degree 3 has at most 3 zeroes by the fundamental theorem.
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