CBSE Class 10 Mathematics Polynomials Assignment Set 08

Question 1. The zero of \(cx + d\), \(c \neq 0\) is
(a) \(-\frac{d}{c}\)
(b) \(\frac{c}{d}\)
(c) \(\frac{d}{c}\)
(d) \(-\frac{c}{d}\)
Answer: (a) \(-\frac{d}{c}\)
To find the zero of the polynomial, set \(cx + d = 0\).
\(cx = -d\)
\(x = -\frac{d}{c}\)
In simple words: Put the linear polynomial equal to zero and solve for \(x\).

Exam Tip: For any linear polynomial \(ax + b\), the zero is always given by \(-\frac{\text{constant term}}{\text{coefficient of } x}\).

Question 2. If one zero of \(x^2 + 5x + 6\) is \(-3\), the other zero is
(a) \(2$
(b) \(-2$
(c) \(3$
(d) \(-3$
Answer: (b) \(-2\)
Let the polynomial be \(p(x) = x^2 + 5x + 6\).
Factoring the quadratic polynomial:
\(x^2 + 3x + 2x + 6 = 0\)
\(x(x + 3) + 2(x + 3) = 0\)
\((x + 3)(x + 2) = 0\)
The zeroes are \(-3\) and \(-2\). Since one zero is given as \(-3\), the other zero is \(-2\).
Alternatively, using the sum of zeroes: \(\alpha + \beta = -\frac{b}{a} = -5\).
\(-3 + \beta = -5 \implies \beta = -2\).
In simple words: Find the roots by splitting the middle term or by using the relation between coefficients and zeroes.

Exam Tip: Using the sum or product of roots is usually faster than full factorization when one root is already known.

Question 3. The sum of zeroes in polynomial \(4x^3 - 3x^2 + 2x + 5\) is
(a) \(\frac{4}{3}\)
(b) \(\frac{3}{4}\)
(c) \(-\frac{3}{4}\)
(d) \(-\frac{4}{3}\)
Answer: (b) \(\frac{3}{4}\)
For a cubic polynomial \(ax^3 + bx^2 + cx + d\), the sum of zeroes is given by \(-\frac{b}{a}\).
Here, \(a = 4\), \(b = -3\), \(c = 2\), and \(d = 5\).
Sum of zeroes = \(-\frac{-3}{4} = \frac{3}{4}\).
In simple words: Take the negative of the coefficient of \(x^2\) and divide it by the coefficient of \(x^3\).

Exam Tip: Remember that the formula for the sum of zeroes is always \(-\frac{b}{a}\), regardless of whether it is a quadratic or cubic polynomial.

Question 4. The sum and product of zeroes of a polynomial are \(\frac{1}{4}\) and \(\frac{1}{2}\), the polynomial is
(a) \(4x^2 - x + 2$
(b) \(x^2 - 4x + 2$
(c) \(4x^2 - 2x + 1$
(d) None of the options
Answer: (a) \(4x^2 - x + 2\)
A quadratic polynomial with sum of zeroes \(S\) and product of zeroes \(P\) is given by \(k(x^2 - Sx + P)\).
Given \(S = \frac{1}{4}\) and \(P = \frac{1}{2}\).
Polynomial = \(k\left(x^2 - \frac{1}{4}x + \frac{1}{2}\right)\)
For \(k = 4\), the polynomial becomes:
\(4\left(x^2 - \frac{1}{4}x + \frac{1}{2}\right) = 4x^2 - x + 2\).
In simple words: Substitute the sum and product into the standard template \(x^2 - (\text{sum})x + \text{product}\) and scale by the denominator.

Exam Tip: Always multiply through by the least common multiple of the denominators to get integer coefficients.

Question 5. The maximum number of zeroes in a cubic polynomial is
(a) 2
(b) 3
(c) 1
(d) 4
Answer: (b) 3
The maximum number of zeroes of a polynomial is equal to its degree. A cubic polynomial has degree 3, so it can have at most 3 real zeroes.
In simple words: The degree of the polynomial tells you the maximum number of zeroes it can have.

Question 6. If the graph of the polynomial cuts y-axis at two different points then the number of zeroes of polynomial are
(a) 2
(b) 1
(c) 0
(d) 3
Answer: (c) 0
The zeroes of a polynomial \(p(x)\) are the x-coordinates of the points where the graph of \(y = p(x)\) intersects the x-axis. Intersection with the y-axis does not determine the zeroes of \(p(x)\). A polynomial function can intersect the y-axis at only one point, but the premise is theoretically asking about intersections with the axis defining zeroes, or if interpreted strictly, a function cannot intersect the y-axis at two distinct points as it would fail the vertical line test. Thus, the number of zeroes corresponding to x-axis intersections cannot be determined as 2 from y-axis cuts, making 0 the correct count for x-axis intersections based on standard trick questions, or interpreted as standard graph properties where standard polynomial graphs intersect the y-axis exactly once.
In simple words: Zeroes depend strictly on intersections with the x-axis, not the y-axis.

Exam Tip: Always look for intersections with the x-axis when counting zeroes from a graph.

Question 7. The maximum number of terms in a linear polynomial is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2
A linear polynomial has the general form \(ax + b\) where \(a \neq 0\). This expression contains exactly 2 terms.
In simple words: A linear polynomial has an x term and a constant term, making 2 terms in total.

Exam Tip: Degree \(n\) polynomial can have up to \(n+1\) terms.

Question 8. The degree of polynomial \(5x^3 - 3x^2 + 6x + 2\) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3
The degree of a polynomial is the highest power of the variable \(x\) in the polynomial expression. Here, the highest power is 3.
In simple words: Look at the highest exponent of \(x\), which is 3.

Exam Tip: The degree is always the exponent of the leading term when the polynomial is written in standard descending order.

FILL IN THE GAP.

Question 9. Polynomial of degree two is called ________
Answer: quadratic polynomial
In simple words: A polynomial of degree 2 is named a quadratic polynomial.

Exam Tip: Memorize polynomial names by degree: 1 is linear, 2 is quadratic, 3 is cubic, 4 is biquadratic.

Question 10. The zeroes of the polynomial \(p(x)\) are precisely the ________ of the points where the graph \(y = f(x)\) intersects the x axis.
Answer: x-coordinates
In simple words: The x-value where the graph touches or crosses the horizontal x-axis gives the zero.

Exam Tip: Zeroes are always coordinate values along the x-axis.

Question 11. A quadratic polynomial can have at most ________ zeroes and a cubic polynomial can have at most ________ zeroes.
Answer: 2, 3
In simple words: Maximum zeroes equal the degree of the polynomial.

Exam Tip: Quadratic = 2, Cubic = 3.

Question 12. If \(\alpha, \beta\) are the zeroes of the polynomial \(p(x) = ax^2 + bx + c\) then \(\alpha + \beta =\) ________, \(\alpha\beta =\) ________
Answer: \(-\frac{b}{a}\), \(\frac{c}{a}\)
In simple words: Sum of zeroes is \(-b/a\) and product of zeroes is \(c/a\).

Exam Tip: These standard relationships are foundational for solving quadratic root problems.

Question 13. If \(\alpha, \beta, \gamma\) are the zeroes of the polynomial \(p(x) = ax^3 + bx^2 + cx + d\) then
i) \(\alpha + \beta + \gamma =\) ________
ii) \(\alpha\beta + \beta\gamma + \gamma\alpha =\) ________
iii) \(\alpha\beta\gamma =\) ________
Answer:
i) \(-\frac{b}{a}\)
ii) \(\frac{c}{a}\)
iii) \(-\frac{d}{a}\)
In simple words: Cubic polynomials have three relations: sum is \(-b/a\), sum of products taken two at a time is \(c/a\), and product is \(-d/a\).

Exam Tip: Note the alternating signs: negative, positive, negative for coefficients \(b\), \(c\), and \(d\) divided by \(a\).

Question 14. The division algorithm states that for polynomial \(p(x)\) and a non-zero polynomial \(g(x)\), there are two polynomial \(q(x)\) and \(r(x)\) :
\(p(x) =\) ________ \(+\) ________
where \(r(x) = 0\) or degree \(r(x)\) ________ degree \(g(x)\).
Answer: \(g(x) \times q(x)\), \(r(x)\), \(<\) (less than)
Dividend = (Divisor \(\times\) Quotient) + Remainder, where remainder is zero or has degree strictly less than the divisor degree.
In simple words: Polynomial division follows the same rules as regular number division: Dividend = Divisor \(\times\) Quotient + Remainder.

Question 15. A quadratic polynomial in \(x\) with real coefficient is ________
Answer: of the form \(ax^2 + bx + c\) where \(a, b, c\) are real numbers and \(a \neq 0\)
In simple words: Any quadratic expression with real numbers as coefficients.

Exam Tip: Always specify the non-zero condition for the leading coefficient \(a\).