Mathematics Classes VI-X for School Assistants and TET Paper 2A · Chapter 3

Polynomials, Zeroes and Linear Equations in Two Variables

What to remember

A zero of a polynomial p(x) is a value k with p(k) = 0. Geometrically, it is where the graph of y = p(x) meets the x-axis. A polynomial of degree n has at most n zeroes.
For a quadratic ax² + bx + c with zeroes α and β: α + β = −b/a and αβ = c/a. The factor theorem and remainder theorem connect zeroes with division.
A pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 has one solution, no solution or infinitely many solutions. Compare the ratios a₁/a₂, b₁/b₂ and c₁/c₂ to decide which case applies.

Polynomials: basic terms

A polynomial in x is an expression such as a₀ + a₁x + a₂x² + … + aₙxⁿ, where the powers of x are whole numbers and the coefficients are real numbers. The degree is the highest power of x with a non-zero coefficient. Expressions like x^(−1) + 2 or √x + 3 are not polynomials, because the powers are not whole numbers.

Degree	Name	Standard form	Example
0	Constant	k (k ≠ 0)	7
1	Linear	ax + b	3x − 5
2	Quadratic	ax² + bx + c	x² − 5x + 6
3	Cubic	ax³ + bx² + cx + d	x³ − 2x² − x + 2

The zero polynomial has no defined degree. A polynomial with one term is a monomial, with two terms a binomial and with three terms a trinomial. The value of p(x) at x = k is found by putting k in place of x. Example: p(x) = x² − 3x − 4 gives p(4) = 16 − 12 − 4 = 0, so 4 is a zero.

Zeroes and their graphs

The graph of y = p(x) shows the zeroes as the points where it cuts the x-axis.

Polynomial	Graph	Number of zeroes
Linear ax + b (a ≠ 0)	Straight line	Exactly 1, at x = −b/a
Quadratic ax² + bx + c	Parabola, opening upward if a > 0 and downward if a < 0	0, 1 or 2
Cubic	Curve with a bend or two	1, 2 or 3

A parabola that does not meet the x-axis has no real zero. A parabola that touches the x-axis at one point has one repeated zero. The zero of a polynomial is not the same as the zero polynomial; for example, 0 can be a zero of x² − 3x without it being the zero polynomial.

Relation between zeroes and coefficients

For the quadratic p(x) = ax² + bx + c with zeroes α and β:

Sum of zeroes: α + β = −b/a.
Product of zeroes: αβ = c/a.
A quadratic with given sum S and product P is x² − Sx + P (or any non-zero multiple of it).

For the cubic ax³ + bx² + cx + d with zeroes α, β, γ:

α + β + γ = −b/a.
αβ + βγ + γα = c/a.
αβγ = −d/a.

Worked example 1: Zeroes of x² − 5x + 6 are 2 and 3 (since (x − 2)(x − 3)). Sum = 5 = −(−5)/1 and product = 6 = 6/1, so the relation holds.

Worked example 2: Find a quadratic polynomial whose zeroes have sum 4 and product 3. It is x² − 4x + 3.

Worked example 3: If one zero of 2x² − 7x + k is 3, then 18 − 21 + k = 0, so k = 3. The other zero follows from the product: 3 × β = 3/2, so β = 1/2.

Division algorithm, remainder theorem and factor theorem

Division algorithm: for polynomials p(x) and g(x) with g(x) ≠ 0, there are polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).

Remainder theorem: when p(x) is divided by (x − a), the remainder is p(a).

Factor theorem: (x − a) is a factor of p(x) if and only if p(a) = 0.

Examples: The remainder when x³ + 2x² − 5x + 1 is divided by (x − 1) is 1 + 2 − 5 + 1 = −1. For p(x) = x³ − 6x² + 11x − 6, p(1) = 0, so (x − 1) is a factor. For the factor (x + a), evaluate at x = −a.

Useful identities: (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; a² − b² = (a + b)(a − b); (a + b)³ = a³ + 3a²b + 3ab² + b³; a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²).

Linear equations in two variables

An equation that can be written as ax + by + c = 0 (a and b not both zero) is a linear equation in two variables. It has infinitely many solutions, and each solution is an ordered pair (x, y). Its graph is a straight line, and every point on the line is a solution. The line x = k is parallel to the y-axis, and y = k is parallel to the x-axis. A pair of such equations is a system of simultaneous equations.

Consistency of a pair

Condition	Lines	Solutions	Name
a₁/a₂ ≠ b₁/b₂	Intersecting	Exactly one	Consistent, independent
a₁/a₂ = b₁/b₂ = c₁/c₂	Coincident	Infinitely many	Consistent, dependent
a₁/a₂ = b₁/b₂ ≠ c₁/c₂	Parallel	None	Inconsistent

Example: 2x + 3y = 7 and 4x + 6y = 14 have ratios 1/2, 1/2, 1/2, so the lines coincide. And x + 2y = 3 with 2x + 4y = 8 have ratios 1/2, 1/2, 3/8, so the lines are parallel and there is no solution.

Methods of solving

Graphical method: draw both lines; the point of intersection is the solution.
Substitution: write one variable in terms of the other from one equation and put it into the second.
Elimination: multiply the equations so that one variable has equal coefficients, then add or subtract.
Cross-multiplication: for a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, x = (b₁c₂ − b₂c₁)/(a₁b₂ − a₂b₁) and y = (c₁a₂ − c₂a₁)/(a₁b₂ − a₂b₁), provided a₁b₂ − a₂b₁ ≠ 0.

Worked example 4: Solve x + y = 14 and x − y = 4. Adding gives 2x = 18, so x = 9 and y = 5.

Worked example 5: Solve 2x + 3y = 12 and x − y = 1. From the second, x = y + 1. Putting in the first gives 2y + 2 + 3y = 12, so y = 2 and x = 3.

Worked example 6 (word problem): The sum of two numbers is 25 and their difference is 7. Let them be x and y: x + y = 25 and x − y = 7, so x = 16 and y = 9.

Worked example 7: For which value of k does kx + 3y = 3 and 12x + ky = 6 have no solution? Parallel lines need k/12 = 3/k ≠ 3/6, so k² = 36, k = ±6. The case k = 6 gives 6/12 = 3/6, which is the coincident case, so k = −6.

Equations that are not linear can sometimes be reduced to linear form. For 2/x + 3/y = 7 and 1/x − 1/y = 1, put u = 1/x and v = 1/y, solve for u and v, and then invert.

Classroom angle

Let students first plot points of x + y = 5 on a grid and see that the points fall on a line. Then two lines on the same grid show the three cases visually. Common errors: sign mistakes while moving terms and dropping the constant in the cross-multiplication formula. Ask students to always check the answer in both original equations.

Exam traps

Zero versus zero polynomial: a zero is a value of x; the zero polynomial is the constant 0.
Sum of zeroes: it is −b/a, not b/a.
Product of zeroes: c/a for a quadratic, but −d/a for a cubic.
Remainder for (x + a): it is p(−a), not p(a).
Degree: a polynomial like (x² + 1)² has degree 4, not 2.
Parallel and coincident lines: both have equal a/b ratios; only the c ratio differs.
Number of zeroes: a quadratic may have fewer than 2 real zeroes; the graph of the quadratic need not cut the x-axis.
Non-polynomials: 1/x and √x are not polynomials.

One-liners

1. The degree of a polynomial is its highest power of x.
2. A linear polynomial has exactly one zero.
3. A polynomial of degree n has at most n zeroes.
4. The graph of a quadratic polynomial is a parabola.
5. For ax² + bx + c: α + β = −b/a and αβ = c/a.
6. A quadratic with sum S and product P is x² − Sx + P.
7. Remainder theorem: the remainder on dividing by (x − a) is p(a).
8. Factor theorem: (x − a) is a factor if p(a) = 0.
9. p(x) = g(x) q(x) + r(x) is the division algorithm.
10. Intersecting lines give a unique solution.
11. Coincident lines give infinitely many solutions.
12. Parallel lines give no solution.

Practice questions

The degree of the polynomial 3x⁴ + 2x² − 7 is
1. 2
2. 3
3. 7
4. 4
Answer
D. 4
The highest power of x is 4.
The zero of the polynomial 3x − 6 is
1. −2
2. 2
3. 6
4. 3
Answer
B. 2
3x − 6 = 0 gives x = 2.
The zeroes of x² − 5x + 6 are
1. −2 and −3
2. 1 and 6
3. 2 and 3
4. 2 and −3
Answer
C. 2 and 3
x² − 5x + 6 = (x − 2)(x − 3).
The sum of the zeroes of 2x² − 8x + 6 is
1. −4
2. 3
3. 8
4. 4
Answer
D. 4
Sum = −b/a = 8/2 = 4.
The product of the zeroes of 3x² + 5x − 2 is
1. −5/3
2. 2/3
3. 5/3
4. −2/3
Answer
D. −2/3
Product = c/a = −2/3.
A quadratic polynomial whose zeroes have sum −3 and product 2 is
1. x² + 3x − 2
2. x² − 2x − 3
3. x² + 3x + 2
4. x² − 3x + 2
Answer
C. x² + 3x + 2
x² − (sum)x + product = x² + 3x + 2.
The remainder when x³ − 3x² + 2x − 5 is divided by (x − 2) is
1. 5
2. −1
3. 3
4. −5
Answer
D. −5
p(2) = 8 − 12 + 4 − 5 = −5.
If (x − 1) is a factor of x² + kx − 3, then k is
1. −2
2. 3
3. 1
4. 2
Answer
D. 2
p(1) = 1 + k − 3 = 0, so k = 2.
The maximum number of zeroes of a quadratic polynomial is
1. 2
2. 1
3. 3
4. 4
Answer
A. 2
A polynomial of degree n has at most n zeroes.
The sum of the zeroes of x³ − 6x² + 11x − 6 is
1. −6
2. 11
3. 6
4. −11
Answer
C. 6
Sum = −b/a = 6.
Which of the following is a polynomial?
1. x² + 3x + 1
2. x + 1/x
3. x^(−2) + 5
4. √x + 2
Answer
A. x² + 3x + 1
Powers of x must be whole numbers.
The degree of (x² + 1)² is
1. 2
2. 3
3. 5
4. 4
Answer
D. 4
Expanding gives x⁴ + 2x² + 1.
The remainder when p(x) is divided by (x + 3) is
1. p(3)
2. p(0)
3. p(−3)
4. −p(3)
Answer
C. p(−3)
x + 3 = x − (−3), so the remainder is p(−3).
The graph of a linear polynomial ax + b (a ≠ 0) cuts the x-axis at
1. three points
2. two points
3. no point
4. exactly one point
Answer
D. exactly one point
A straight line meets the x-axis once, at x = −b/a.
The graph of y = ax² + bx + c with a > 0 is a parabola that opens
1. upward
2. downward
3. to the right
4. to the left
Answer
A. upward
A positive coefficient of x² opens it upward.
If 2 is a zero of x² − 5x + k, then k is
1. −6
2. 10
3. 3
4. 6
Answer
D. 6
4 − 10 + k = 0, so k = 6.
If α and β are the zeroes of x² − 7x + 12, then α² + β² is
1. 37
2. 49
3. 25
4. 19
Answer
C. 25
α² + β² = (α + β)² − 2αβ = 49 − 24 = 25.
If α and β are the zeroes of x² − 5x + 6, then 1/α + 1/β is
1. 6/5
2. 5/6
3. 1/5
4. 11/6
Answer
B. 5/6
(α + β)/αβ = 5/6.
The solution of x + y = 14 and x − y = 4 is
1. x = 5, y = 9
2. x = 10, y = 4
3. x = 9, y = 5
4. x = 8, y = 6
Answer
C. x = 9, y = 5
Adding gives 2x = 18, so x = 9 and y = 5.
In the pair 2x + 3y = 12 and x − y = 1, the value of y is
1. 3
2. 1
3. −1
4. 2
Answer
D. 2
x = y + 1 gives 5y + 2 = 12, so y = 2 (and x = 3).
The pair 2x + 3y = 7 and 4x + 6y = 14 has
1. infinitely many solutions
2. no solution
3. a unique solution
4. exactly two solutions
Answer
A. infinitely many solutions
The ratios 1/2, 1/2, 1/2 are equal, so the lines coincide.
The pair x + 2y = 3 and 2x + 4y = 8 has
1. exactly two solutions
2. no solution
3. a unique solution
4. infinitely many solutions
Answer
B. no solution
Ratios 1/2 = 1/2 ≠ 3/8, so the lines are parallel.
The solution of 3x + 2y = 5 and x − y = 0 is
1. x = 1, y = 1
2. x = 2, y = 2
3. x = 5, y = 0
4. x = 0, y = 5/2
Answer
A. x = 1, y = 1
Putting y = x gives 5x = 5.
The value of k for which kx + 3y = 3 and 12x + ky = 6 have no solution is
1. 12
2. −12
3. −6
4. 6
Answer
C. −6
k/12 = 3/k gives k = ±6; k = 6 makes the lines coincide, so only k = −6 gives parallel lines.
The graph of a linear equation in two variables is
1. a straight line
2. a circle
3. a single point
4. a parabola
Answer
A. a straight line
Every solution lies on a line.
A linear equation in two variables has
1. exactly two solutions
2. infinitely many solutions
3. no solution
4. exactly one solution
Answer
B. infinitely many solutions
Each point on its line is a solution.
The sum of two numbers is 25 and their difference is 7. The larger number is
1. 15
2. 9
3. 16
4. 18
Answer
C. 16
x + y = 25 and x − y = 7 give x = 16.
The sum of the ages of a father and son is 40 years, and the father is 3 times as old as the son. The son's age is
1. 8 years
2. 12 years
3. 13 years
4. 10 years
Answer
D. 10 years
x + y = 40 and x = 3y give 4y = 40.
The graph of the equation x = 3 is a line
1. parallel to the y-axis
2. parallel to the x-axis
3. through the origin
4. at 45° to both axes
Answer
A. parallel to the y-axis
Every point on it has x = 3.
On putting u = 1/x and v = 1/y, the pair 2/x + 3/y = 7 and 1/x − 1/y = 1 gives x equal to
1. 2
2. 1/2
3. 1
4. 3
Answer
B. 1/2
2u + 3v = 7 and u − v = 1 give u = 2, v = 1, so x = 1/2.
Which is a factor of x³ − 6x² + 11x − 6?
1. x + 1
2. x + 6
3. x − 4
4. x − 1
Answer
D. x − 1
p(1) = 1 − 6 + 11 − 6 = 0.
If 3 is a zero of 2x² − 7x + k, then the other zero is
1. 3/2
2. 2
3. −1/2
4. 1/2
Answer
D. 1/2
k = 3; the product is 3/2, so the other zero is 1/2.
A pair of linear equations has a unique solution when
1. a₁/a₂ = b₁/b₂ = c₁/c₂
2. a₁/a₂ = b₁/b₂ ≠ c₁/c₂
3. a₁/a₂ ≠ b₁/b₂
4. a₁/a₂ = b₁/b₂
Answer
C. a₁/a₂ ≠ b₁/b₂
Unequal a and b ratios mean intersecting lines.
If a + b = 5 and ab = 6, then a² + b² is
1. 13
2. 30
3. 25
4. 11
Answer
A. 13
a² + b² = 25 − 12 = 13.
The solution of 2x − y = 4 and x + y = 5 is
1. (2, 3)
2. (3, 2)
3. (4, 1)
4. (1, 4)
Answer
B. (3, 2)
Adding gives 3x = 9, so x = 3 and y = 2.
Consider the statements: 1. The degree of a polynomial is the highest power of its variable. 2. Every quadratic polynomial has exactly two real zeroes.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
A quadratic can have 0, 1 or 2 real zeroes.
Consider the statements: 1. For ax² + bx + c, the sum of zeroes is −b/a. 2. For ax² + bx + c, the product of zeroes is c/a.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both relations are standard.
Consider the statements: 1. When p(x) is divided by (x + a), the remainder is p(a). 2. (x − a) is a factor of p(x) if p(a) = 0.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
The remainder for (x + a) is p(−a); the factor theorem in statement 2 is correct.
Consider the statements: 1. Intersecting lines give a unique solution. 2. Parallel lines give infinitely many solutions.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Parallel lines never meet, so there is no solution.
Consider the statements: 1. Coincident lines give infinitely many solutions. 2. Parallel lines give no solution.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
These are the two special cases of a consistent-dependent and an inconsistent pair.
Consider the statements: 1. 1/x + 2 is a polynomial. 2. A linear polynomial has exactly one zero.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
1/x has power −1, so it is not a polynomial.
Consider the statements: 1. A cubic polynomial has at most 3 zeroes. 2. The graph of a cubic polynomial is a parabola.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
The parabola is the graph of a quadratic polynomial.
Consider the statements: 1. A cubic polynomial always has exactly 3 real zeroes. 2. The graph of the equation y = 4 is parallel to the y-axis.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
D. Neither 1 nor 2
A cubic can have 1, 2 or 3 real zeroes; y = 4 is parallel to the x-axis.
Consider the statements: 1. The product of the zeroes of ax³ + bx² + cx + d is −d/a. 2. The sum of the zeroes of ax³ + bx² + cx + d is c/a.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
The sum is −b/a; c/a is the sum of products taken two at a time.
The expression a² − b² factorises as
1. (a + b)²
2. a(a − b)
3. (a − b)²
4. (a + b)(a − b)
Answer
D. (a + b)(a − b)
Difference of squares identity.

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1.For what positive values of k do the following pair of linear equations have infinitely many solutions? kx + 3y – ( k – 3 ) = 0 12x + ky – k = 0
2.Given a linear equation in two variables: 5x + 7y - 8 = 0, which of the following linear equations, along with the given equation, forms a system of linear equations having no solution?
3.What is the area of the region created by the two linear equations 2x + y = 6 and 2x – y = –2 bounded by these lines and the x-axis?
4.The product of two consecutive positive integers is 247. If the smaller of these two integers is represented by x, which of the following will correspond to the equation for finding out the value of x?
5.Let a be an odd positive integer. If the roots of a quadratic equation x² − (a − 1)x + 4a + 7 = 0 are integers, then the least value of a is:
6.What should come in place of the question mark (?) in the following equation?
7.If the intersection point of the equations 3p − 2q + 23 = 0 and −2p + q − 13 = 0 on the graph is (p, q), then the value of (3p + q) is equal to:
8.A batsman scores 90 in his 16th innings, raising his average from 42 to 45. Which equation confirms his 16th-innings score?
9.What will come in place of ‘?’ in the following equation, if ‘+’ and ‘-‘ are interchanged and ‘×’ and ‘÷’ are interchanged? 95 ÷ 5 + 378 × 3 - 161 = ?
10.What will come in the place of ‘?’ in the following equation, if ‘+’ and ‘-‘ are interchanged and ‘×’ and ‘÷’ are interchanged? 216 ÷ 3 + 1313 × 13 - 112 = ?

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