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Q1(iii):
Find the value of the polynomial $5x – 4x^2 + 3$ at
(iii) $x = 2$
Solution :
Given Variables & Initial Setup
We are given a polynomial in a single variable, $x$. Let us define the polynomial as a function $P(x)$:
$P(x) = 5x - 4x^2 + 3$
The objective is to evaluate this polynomial at the specific domain value $x = 2$. [Per the fundamental theorem of polynomial evaluation, finding the value of a polynomial at a given point requires the direct substitution of that point into the variable, followed by simplification according to the standard order of operations (PEMDAS/BODMAS)].
Step 1: Substitution of the Variable
Substitute $x = 2$ into every instance of $x$ within the polynomial $P(x)$:
$P(2) = 5(2) - 4(2)^2 + 3$
Step 2: Resolution of Exponents
According to the order of operations, exponentiation must be resolved before multiplication. We evaluate the quadratic term $(2)^2$:
$2^2 = 2 \times 2 = 4$
Substituting this back into the equation yields:
$P(2) = 5(2) - 4(4) + 3$
Step 3: Execution of Multiplication
Next, perform the scalar multiplications for both the linear and quadratic terms:
- Linear term: $5 \times 2 = 10$
- Quadratic term: $4 \times 4 = 16$
Updating the polynomial expression:
$P(2) = 10 - 16 + 3$
Step 4: Sequential Addition and Subtraction
Finally, evaluate the arithmetic expression from left to right:
$P(2) = (10 - 16) + 3$
$P(2) = -6 + 3$
$P(2) = -3$
Graphical Verification
The polynomial $P(x) = -4x^2 + 5x + 3$ represents a downward-opening parabola [since the leading coefficient $a = -4$ is less than zero]. Evaluating the polynomial at $x = 2$ corresponds to finding the $y$-coordinate of the point on this parabola where the $x$-coordinate is $2$. As calculated, this point exists exactly at $(2, -3)$ on the Cartesian plane.
Final Solution: The value of the polynomial $5x - 4x^2 + 3$ at $x = 2$ is $-3$.
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) $x = –1$
- Q2(i): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (i) $p(y) = y^2 – y + 1$
- 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
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