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Q2(i):
Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials:
(i) $p(y) = y^2 – y + 1$
Solution :
Given Polynomial & Initial Setup
We are given the quadratic polynomial in terms of the variable $y$:
$p(y) = y^2 - y + 1$
To find the values of $p(0)$, $p(1)$, and $p(2)$, we must systematically substitute the given numerical values into the independent variable $y$ of the polynomial. [Per the Fundamental Theorem of Polynomial Evaluation, substituting a constant $c$ into $p(y)$ yields the value of the function at that specific coordinate].
Step 1: Evaluating $p(0)$
Substitute $y = 0$ into the polynomial equation:
$p(0) = (0)^2 - (0) + 1$
$p(0) = 0 - 0 + 1$
$p(0) = 1$
[Justification: The zero property of multiplication dictates that any power or multiple of zero is zero, leaving only the constant term].
Step 2: Evaluating $p(1)$
Substitute $y = 1$ into the polynomial equation:
$p(1) = (1)^2 - (1) + 1$
$p(1) = 1 - 1 + 1$
$p(1) = 0 + 1$
$p(1) = 1$
[Justification: The additive inverse property ensures that $1 - 1 = 0$, simplifying the expression to the constant $1$].
Step 3: Evaluating $p(2)$
Substitute $y = 2$ into the polynomial equation:
$p(2) = (2)^2 - (2) + 1$
$p(2) = 4 - 2 + 1$
$p(2) = 2 + 1$
$p(2) = 3$
[Justification: Following the standard order of operations (PEMDAS/BODMAS), we first evaluate the exponent $(2^2 = 4)$, then perform sequential addition and subtraction from left to right].
Graphical Representation of the Polynomial
The polynomial $p(y) = y^2 - y + 1$ forms a parabola. The points we just evaluated—$(0, 1)$, $(1, 1)$, and $(2, 3)$—represent exact coordinates on this curve. Below is the precise geometric plot of the function.
Final Solution: The evaluated values for the polynomial are $p(0) = 1$, $p(1) = 1$, and $p(2) = 3$.
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.2
- Q1(i): Find the value of the polynomial $5x – 4x^2 + 3$ at (i) $x = 0$
- Q1(ii): Find the value of the polynomial $5x – 4x^2 + 3$ at (ii) $x = –1$
- Q1(iii): Find the value of the polynomial $5x – 4x^2 + 3$ at (iii) $x = 2$
- Q2(ii): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (ii) $p(t) = 2 + t + 2t^2 – t^3$
- Q2(iii): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (iii) $p(x) = x^3$
- Q2(iv): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (iv) $p(x) = (x – 1) (x + 1)$
- Q3(i): Verify whether the following are zeroes of the polynomial, indicated against them. (i) $p(x) = 3x + 1, x = –\frac{1}{3}$
- Q3(ii): Verify whether the following are zeroes of the polynomial, indicated against them. (ii) $p(x) = 5x – \pi, x = \frac{4}{5}$
- Q3(iii): Verify whether the following are zeroes of the polynomial, indicated against them. (iii) $p(x) = x^2 – 1, x = 1, –1$
- Q3(iv): Verify whether the following are zeroes of the polynomial, indicated against them. (iv) $p(x) = (x + 1) (x – 2), x = – 1, 2$
- Q3(v): Verify whether the following are zeroes of the polynomial, indicated against them. (v) $p(x) = x^2, x = 0$
- Q3(vi): Verify whether the following are zeroes of the polynomial, indicated against them. (vi) $p(x) = lx + m, x = –\frac{m}{l}$
- Q3(vii): Verify whether the following are zeroes of the polynomial, indicated against them. (vii) $p(x) = 3x^2 – 1, x = –\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}$
- Q3(viii): Verify whether the following are zeroes of the polynomial, indicated against them. (viii) $p(x) = 2x + 1, x = \frac{1}{2}$
- Q4(i): Find the zero of the polynomial in each of the following cases: (i) $p(x) = x + 5$
- Q4(ii): Find the zero of the polynomial in each of the following cases: (ii) $p(x) = x – 5$
- Q4(iii): Find the zero of the polynomial in each of the following cases: (iii) $p(x) = 2x + 5$
- Q4(iv): Find the zero of the polynomial in each of the following cases: (iv) $p(x) = 3x – 2$
- Q4(v): Find the zero of the polynomial in each of the following cases: (v) $p(x) = 3x$
- Q4(vi): Find the zero of the polynomial in each of the following cases: (vi) $p(x) = ax, a \neq 0$
- Q4(vii): Find the zero of the polynomial in each of the following cases: (vii) $p(x) = cx + d, c \neq 0, c, d$ are real numbers.
CBSE Solutions for Class 9 Mathematics Polynomials
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Polynomials
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